Class 12 Math Ch-7 Integration MCQs Exam 2027 
Class 12 Math Ch-7 Integration MCQs Exam 2027 Details: नीचे दिए गए सभी Questions Bihar Board परीक्षा 2027 के लिए “Very Very Important Multiple Choice Questions (MCQs) Objective” (अत्यंत महत्वपूर्ण प्रश्न) हैं। इन सभी Class 12th के (Mathematics/गणित) = गणित भाग-2 (English Medium) Book Chapter-7 Integration का Questions का Solve का वीडियो Youtube और Website पर Upload किया है।
Topic 1: Power Rules and Algebraic Integration (Basic)
$\int x^5 dx$
(A) $\frac{x^6}{6} + K$
(B) $5x^4 + K$
(C) $x^6 + K$
(D) $\frac{x^5}{5} + K$
$\int dx$
(A) $x + K$
(B) $K$
(C) 0
(D) 1
$\int \sqrt{x} dx$
(A) $\frac{2}{3}x^{3/2} + K$
(B) $\frac{3}{2}x^{3/2} + K$
(C) $\frac{2}{3}x^{1/2} + K$
(D) $x^{3/2} + K$
$\int \frac{dx}{\sqrt{x}}$
(A) $2\sqrt{x} + K$
(B) $\frac{\sqrt{x}}{2} + K$
(C) $\sqrt{x} + K$
(D) $2x + K$
$\int x^2 dx$
(A) $\frac{x^3}{3} + K$
(B) $2x + K$
(C) $x^3 + K$
(D) $\frac{x^2}{2} + K$
$\int x^8 dx$
(A) $\frac{x^9}{9} + K$
(B) $8x^7 + K$
(C) $\frac{x^8}{8} + K$
(D) $x^9 + K$
$\int x^4 dx$
(A) $\frac{x^5}{5} + K$
(B) $4x^3 + K$
(C) $x^5 + K$
(D) $5x^5 + K$
$\int x^7 dx$
(A) $\frac{x^8}{8} + K$
(B) $7x^6 + K$
(C) $\frac{x^7}{7} + K$
(D) $x^8 + K$
$\int \frac{1}{x^2} dx$
(A) $-\frac{1}{x} + K$
(B) $\frac{1}{x} + K$
(C) $\log x^2 + K$
(D) $-\frac{2}{x^3} + K$
$\int (x + 2) dx$
(A) $\frac{x^2}{2} + 2x + K$
(B) $(x+2)^2 + K$
(C) $x^2 + 2x + K$
(D) $\log(x+2) + K$
$\int (3x^2 + 4x + 5) dx$
(A) $x^3 + 2x^2 + 5x + K$
(B) $3x^3 + 4x^2 + 5x + K$
(C) $6x + 4 + K$
(D) $x^3 + x^2 + x + K$
$\int \frac{1}{x^3} dx$
(A) $-\frac{1}{2x^2} + K$
(B) $\frac{1}{2x^2} + K$
(C) $-\frac{3}{x^4} + K$
(D) $\log |x^3| + K$
$\int x^{1/2} dx$
(A) $\frac{2}{3}x^{3/2} + K$
(B) $\frac{3}{2}x^{3/2} + K$
(C) $x^{3/2} + K$
(D) $\frac{1}{2}x^{-1/2} + K$
$\int 2x dx$
(A) $x^2 + K$
(B) $2x^2 + K$
(C) $2 + K$
(D) $x + K$
$\int (ax^2 + bx + c) dx$
(A) $\frac{ax^3}{3} + \frac{bx^2}{2} + cx + K$
(B) $ax^3 + bx^2 + cx + K$
(C) $2ax + b + K$
(D) $x^3 + x^2 + x + K$
$\int \frac{dx}{x}$
(A) $\log |x| + K$
(B) $-\frac{1}{x^2} + K$
(C) $e^x + K$
(D) $x + K$
$\int \frac{dx}{x-1}$
(A) $\log |x-1| + K$
(B) $\frac{1}{(x-1)^2} + K$
(C) $(x-1)^2 + K$
(D) $\log |x| + K$
$\int \frac{dx}{x+5}$
(A) $\log |x+5| + K$
(B) $\frac{1}{x+5} + K$
(C) $x+5 + K$
(D) $\log x + 5 + K$
$\int \frac{3 dx}{x}$
(A) $3 \log |x| + K$
(B) $\log |3x| + K$
(C) $\frac{3}{x^2} + K$
(D) $3x + K$
$\int \frac{dx}{2x+3}$
(A) $\frac{1}{2} \log |2x+3| + K$
(B) $\log |2x+3| + K$
(C) $2 \log |2x+3| + K$
(D) $\frac{1}{3} \log |2x+3| + K$
Topic 2: Trigonometric Integration
$\int \sin x dx$
(A) $-\cos x + K$
(B) $\cos x + K$
(C) $\sin x + K$
(D) $-\sin x + K$
$\int \cos x dx$
(A) $\sin x + K$
(B) $-\sin x + K$
(C) $\cos x + K$
(D) $-\cos x + K$
$\int \sec^2 x dx$
(A) $\tan x + K$
(B) $-\tan x + K$
(C) $\cot x + K$
(D) $\sec x + K$
$\int \csc^2 x dx$
(A) $-\cot x + K$
(B) $\cot x + K$
(C) $\tan x + K$
(D) $-\csc x + K$
$\int \sec x \tan x dx$
(A) $\sec x + K$
(B) $\tan x + K$
(C) $\sec^2 x + K$
(D) $-\sec x + K$
$\int \csc x \cot x dx$
(A) $-\csc x + K$
(B) $\csc x + K$
(C) $\cot x + K$
(D) $-\cot x + K$
$\int \sin 2x dx$
(A) $-\frac{\cos 2x}{2} + K$
(B) $\frac{\cos 2x}{2} + K$
(C) $2 \cos 2x + K$
(D) $-\cos 2x + K$
$\int \cos 3x dx$
(A) $\frac{\sin 3x}{3} + K$
(B) $-\frac{\sin 3x}{3} + K$
(C) $3 \sin 3x + K$
(D) $\sin 3x + K$
$\int \sec^2 4x dx$
(A) $\frac{\tan 4x}{4} + K$
(B) $4 \tan 4x + K$
(C) $\tan 4x + K$
(D) $-\frac{\tan 4x}{4} + K$
$\int \cos 6\theta d\theta$
(A) $\frac{\sin 6\theta}{6} + K$
(B) $-\frac{\sin 6\theta}{6} + K$
(C) $6 \sin 6\theta + K$
(D) $\sin 6\theta + K$
The value of $\int \tan x dx$ is:
(A) $\log |\sec x| + K$
(B) $\log |\cos x| + K$
(C) $\sec^2 x + K$
(D) $-\log |\sec x| + K$
$\int \cot x dx$ is equal to:
(A) $\log |\sin x| + K$
(B) $\log |\cos x| + K$
(C) $-\csc^2 x + K$
(D) $\log |\tan x| + K$
$\int \sec x dx$ is equal to:
(A) $\log |\sec x + \tan x| + K$
(B) $\log |\sec x – \tan x| + K$
(C) $\tan x + K$
(D) $\sec x \tan x + K$
The value of $\int \csc x dx$ is:
(A) $\log |\csc x – \cot x| + K$
(B) $\log |\csc x + \cot x| + K$
(C) $-\cot x + K$
(D) $\log |\sin x| + K$
$\int \sin^2 x dx + \int \cos^2 x dx$ is:
(A) $x + K$
(B) $1 + K$
(C) $0 + K$
(D) $2x + K$
The value of $\int \tan^2 x dx$ is:
(A) $\tan x – x + K$
(B) $\tan x + x + K$
(C) $\sec x + K$
(D) $\tan x + K$
$\int \cot^2 x dx$ is equal to:
(A) $-\cot x – x + K$
(B) $\cot x + x + K$
(C) $-\cot x + x + K$
(D) $\csc x + K$
What is the value of $\int \sin^2 x dx$?
(A) $\frac{x}{2} – \frac{\sin 2x}{4} + K$
(B) $\frac{x}{2} + \frac{\sin 2x}{4} + K$
(C) $-\cos^2 x + K$
(D) $\frac{\sin^3 x}{3} + K$
What is the value of $\int \cos^2 x dx$?
(A) $\frac{x}{2} + \frac{\sin 2x}{4} + K$
(B) $\frac{x}{2} – \frac{\sin 2x}{4} + K$
(C) $\sin^2 x + K$
(D) $\frac{\cos^3 x}{3} + K$
$\int \frac{dx}{\sin^2 x \cos^2 x}$ is equal to:
(A) $\tan x – \cot x + K$
(B) $\tan x + \cot x + K$
(C) $\sin x + \cos x + K$
(D) $\tan x \cot x + K$
The value of $\int \frac{\cos 2x – \cos 2\alpha}{\cos x – \cos \alpha} dx$ is:
(A) $2(\sin x + x \cos \alpha) + K$
(B) $2(\sin x – \sin \alpha) + K$
(C) $2(\cos x + x \sin \alpha) + K$
(D) 0
$\int \sqrt{1 + \sin 2x} dx$ is equal to:
(A) $\sin x – \cos x + K$
(B) $\sin x + \cos x + K$
(C) $\cos x – \sin x + K$
(D) $x + K$
The value of $\int \frac{\sin x}{1 + \sin x} dx$ is:
(A) $x – \tan x + \sec x + K$
(B) $x + \tan x – \sec x + K$
(C) $\tan x – x + K$
(D) $\sec x + K$
$\int \frac{1}{1 + \cos x} dx$ is equal to:
(A) $\tan(x/2) + K$
(B) $\cot(x/2) + K$
(C) $\sec(x/2) + K$
(D) $\tan x + K$
The value of $\int \frac{1}{1 – \cos x} dx$ is:
(A) $-\cot(x/2) + K$
(B) $\tan(x/2) + K$
(C) $\csc x + K$
(D) $\cot x + K$
$\int \sin 3x \cos 4x dx$ is equal to:
(A) $-\frac{1}{2} [\frac{\cos 7x}{7} – \cos x] + K$
(B) $\frac{1}{2} [\frac{\sin 7x}{7} + \sin x] + K$
(C) $\cos 7x + \cos x + K$
(D) $-\cos 3x \sin 4x + K$
The value of $\int \cos^3 x dx$ is:
(A) $\sin x – \frac{\sin^3 x}{3} + K$
(B) $\frac{\cos^4 x}{4} + K$
(C) $\sin x + \frac{\sin^3 x}{3} + K$
(D) $-\sin x + K$
$\int \frac{\cos x – \sin x}{\cos x + \sin x} dx$ is equal to:
(A) $\log |\cos x + \sin x| + K$
(B) $\log |\cos x – \sin x| + K$
(C) $\tan x + K$
(D) $e^x + K$
The value of $\int \frac{\sec^2 x}{3 + 4 \tan x} dx$ is:
(A) $\frac{1}{4} \log |3 + 4 \tan x| + K$
(B) $\log |3 + 4 \tan x| + K$
(C) $4 \log |3 + 4 \tan x| + K$
(D) $\tan^{-1} x + K$
$\int \frac{dx}{1 + \tan x}$ is equal to:
(A) $\frac{x}{2} + \frac{1}{2} \log |\cos x + \sin x| + K$
(B) $\log |1 + \tan x| + K$
(C) $\frac{x}{2} – \frac{1}{2} \log |\cos x + \sin x| + K$
(D) $\tan^{-1} x + K$
Topic 3: Exponential and Logarithmic Functions
The value of $\int e^x dx$ is:
(A) $e^x + K$
(B) $e^{-x} + K$
(C) $xe^x + K$
(D) $\frac{e^x}{x} + K$
$\int e^{3x} dx$ is equal to:
(A) $\frac{e^{3x}}{3} + K$
(B) $3e^{3x} + K$
(C) $e^{3x} + K$
(D) $e^x + K$
The value of $\int a^x dx$ is:
(A) $\frac{a^x}{\log a} + K$
(B) $a^x \log a + K$
(C) $a^x + K$
(D) $\frac{\log a}{a^x} + K$
$\int 2^x dx$ is equal to:
(A) $\frac{2^x}{\log 2} + K$
(B) $2^x \log 2 + K$
(C) $2^{x+1} + K$
(D) $\frac{2^x}{x} + K$
The value of $\int e^{ax} dx$ is:
(A) $\frac{e^{ax}}{a} + K$
(B) $ae^{ax} + K$
(C) $e^{ax} + K$
(D) $e^x + K$
$\int e^{x/2} dx$ is equal to:
(A) $2e^{x/2} + K$
(B) $\frac{1}{2}e^{x/2} + K$
(C) $e^{x/2} + K$
(D) $e^x + K$
The value of $\int 5^x dx$ is:
(A) $\frac{5^x}{\log 5} + K$
(B) $5^x \log 5 + K$
(C) $5^x + K$
(D) $\frac{5^{x+1}}{x+1} + K$
$\int e^{\cos x} \sin x dx$ is equal to:
(A) $-e^{\cos x} + K$
(B) $e^{\cos x} + K$
(C) $e^{\sin x} + K$
(D) $\sin x + K$
The value of $\int \frac{e^x}{e^x + 1} dx$ is:
(A) $\log |e^x + 1| + K$
(B) $e^x + K$
(C) $\frac{1}{e^x+1} + K$
(D) $\log e^x + K$
$\int \log 2 dx$ is equal to:
(A) $x \log 2 + K$
(B) $\log 2 + K$
(C) $\frac{1}{2} + K$
(D) 0
The value of $\int \log x dx$ is:
(A) $x \log x – x + K$
(B) $\frac{1}{x} + K$
(C) $x \log x + x + K$
(D) $\log x + K$
$\int \frac{1}{x \log x} dx$ is equal to:
(A) $\log |\log x| + K$
(B) $\log x + K$
(C) $(\log x)^2 + K$
(D) $\frac{1}{\log x} + K$
The value of $\int \frac{e^{\tan^{-1} x}}{1+x^2} dx$ is:
(A) $e^{\tan^{-1} x} + K$
(B) $\tan^{-1} x + K$
(C) $e^x + K$
(D) $\frac{1}{1+x^2} + K$
$\int e^x (x+1) dx$ is equal to:
(A) $x e^x + K$
(B) $e^x + K$
(C) $(x+1)e^x + K$
(D) $x^2 e^x + K$
The value of $\int \frac{e^{1/x}}{x^2} dx$ is:
(A) $-e^{1/x} + K$
(B) $e^{1/x} + K$
(C) $\frac{1}{x} e^{1/x} + K$
(D) $\log e^{1/x} + K$
What will be the value of $\int_0^1 e^x dx$?
(A) $e – 1$
(B) $e$
(C) $1$
(D) $e + 1$
$\int 10^x dx$ is equal to:
(A) $\frac{10^x}{\log 10} + K$
(B) $10^x \log 10 + K$
(C) $10^x + K$
(D) $x \cdot 10^{x-1} + K$
The value of $\int e^{-x} dx$ is:
(A) $-e^{-x} + K$
(B) $e^{-x} + K$
(C) $e^x + K$
(D) $-e^x + K$
$\int \frac{dx}{x(\log x)^2}$ is equal to:
(A) $-\frac{1}{\log x} + K$
(B) $\frac{1}{\log x} + K$
(C) $\log (\log x) + K$
(D) $x \log x + K$
The value of $\int \frac{e^x – e^{-x}}{e^x + e^{-x}} dx$ is:
(A) $\log |e^x + e^{-x}| + K$
(B) $\log |e^x – e^{-x}| + K$
(C) $e^x + e^{-x} + K$
(D) $x + K$
Topic 4: Special Integrals and Inverse Trigonometric Forms
The value of $\int \frac{dx}{x^2 + a^2}$ is:
(A) $\frac{1}{a} \tan^{-1} \frac{x}{a} + K$
(B) $\tan^{-1} \frac{x}{a} + K$
(C) $\frac{1}{a} \sin^{-1} \frac{x}{a} + K$
(D) $\frac{1}{a^2} \tan^{-1} \frac{x}{a} + K$
$\int \frac{dx}{x^2 + 4}$ is equal to:
(A) $\frac{1}{2} \tan^{-1} \frac{x}{2} + K$
(B) $\frac{x}{2} + K$
(C) $\frac{1}{4} \tan^{-1} \frac{x}{4} + K$
(D) $\tan^{-1} \frac{x}{2} + K$
The value of $\int \frac{dx}{\sqrt{1-x^2}}$ is:
(A) $\sin^{-1} x + K$
(B) $\cos^{-1} x + K$
(C) $\tan^{-1} x + K$
(D) $\log |x| + K$
$\int \frac{dx}{1+x^2}$ is equal to:
(A) $\tan^{-1} x + K$
(B) $\sin^{-1} x + K$
(C) $\cot^{-1} x + K$
(D) $\cos^{-1} x + K$
What is the value of $\int \frac{dx}{x^2 – a^2}$?
(A) $\frac{1}{2a} \log |\frac{x-a}{x+a}| + K$
(B) $\frac{1}{2a} \log |\frac{x+a}{x-a}| + K$
(C) $\frac{1}{a} \tan^{-1} \frac{x}{a} + K$
(D) $\log |x^2-a^2| + K$
$\int \frac{dx}{a^2 – x^2}$ is equal to:
(A) $\frac{1}{2a} \log |\frac{a+x}{a-x}| + K$
(B) $\frac{1}{2a} \log |\frac{a-x}{a+x}| + K$
(C) $\frac{1}{a} \sin^{-1} \frac{x}{a} + K$
(D) $\log |a^2-x^2| + K$
The value of $\int \frac{dx}{\sqrt{x^2+a^2}}$ is:
(A) $\log |x + \sqrt{x^2+a^2}| + K$
(B) $\sin^{-1} \frac{x}{a} + K$
(C) $\tan^{-1} \frac{x}{a} + K$
(D) $\frac{1}{a} \log |x + \sqrt{x^2+a^2}| + K$
$\int \frac{dx}{\sqrt{x^2-a^2}}$ is equal to:
(A) $\log |x + \sqrt{x^2-a^2}| + K$
(B) $\frac{1}{a} \sec^{-1} \frac{x}{a} + K$
(C) $\sin^{-1} \frac{x}{a} + K$
(D) $x + \sqrt{x^2-a^2} + K$
The value of $\int \frac{dx}{\sqrt{a^2-x^2}}$ is:
(A) $\sin^{-1} \frac{x}{a} + K$
(B) $\frac{1}{a} \sin^{-1} \frac{x}{a} + K$
(C) $\tan^{-1} \frac{x}{a} + K$
(D) $\cos^{-1} \frac{x}{a} + K$
$\int \frac{dx}{x \sqrt{x^2-1}}$ is equal to:
(A) $\sec^{-1} x + K$
(B) $\csc^{-1} x + K$
(C) $\sin^{-1} x + K$
(D) $\tan^{-1} x + K$
The value of $\int \frac{dx}{x^2+9}$ is:
(A) $\frac{1}{3} \tan^{-1} \frac{x}{3} + K$
(B) $\tan^{-1} \frac{x}{3} + K$
(C) $\frac{1}{3} \sin^{-1} \frac{x}{3} + K$
(D) $\frac{1}{9} \tan^{-1} \frac{x}{9} + K$
$\int \frac{dx}{x^2-16}$ is equal to:
(A) $\frac{1}{8} \log |\frac{x-4}{x+4}| + K$
(B) $\frac{1}{4} \log |\frac{x-4}{x+4}| + K$
(C) $\frac{1}{8} \log |\frac{x+4}{x-4}| + K$
(D) $\frac{1}{2} \tan^{-1} \frac{x}{4} + K$
The value of $\int \frac{dx}{\sqrt{4-x^2}}$ is:
(A) $\sin^{-1} \frac{x}{2} + K$
(B) $\frac{1}{2} \sin^{-1} \frac{x}{2} + K$
(C) $\tan^{-1} \frac{x}{2} + K$
(D) $\cos^{-1} \frac{x}{2} + K$
$\int \frac{x^2}{1+x^6} dx$ is equal to:
(A) $\frac{1}{3} \tan^{-1} (x^3) + K$
(B) $\tan^{-1} (x^3) + K$
(C) $\frac{1}{3} \tan^{-1} x + K$
(D) $\log |1+x^6| + K$
The value of $\int \frac{dx}{x^2+2x+2}$ is:
(A) $\tan^{-1} (x+1) + K$
(B) $\tan^{-1} x + 1 + K$
(C) $\log |x^2+2x+2| + K$
(D) $\sin^{-1} (x+1) + K$
$\int \frac{dx}{\sqrt{9-25x^2}}$ is equal to:
(A) $\frac{1}{5} \sin^{-1} \frac{5x}{3} + K$
(B) $\sin^{-1} \frac{5x}{3} + K$
(C) $\frac{1}{3} \sin^{-1} \frac{5x}{3} + K$
(D) $\frac{1}{5} \cos^{-1} \frac{5x}{3} + K$
The value of $\int \frac{dx}{x^2-x}$ is:
(A) $\log |\frac{x-1}{x}| + K$
(B) $\log |\frac{x}{x-1}| + K$
(C) $\log |x^2-x| + K$
(D) $\frac{1}{x} \log x + K$
$\int \frac{dx}{x^2+x}$ is equal to:
(A) $\log |\frac{x}{x+1}| + K$
(B) $\log |\frac{x+1}{x}| + K$
(C) $\log |x^2+x| + K$
(D) 0
The value of $\int \frac{2x}{x^2+1} dx$ is:
(A) $\log |x^2+1| + K$
(B) $\tan^{-1} x + K$
(C) $2 \log |x^2+1| + K$
(D) $x^2 + K$
$\int \frac{1}{x^2+2x+5} dx$ is equal to:
(A) $\frac{1}{2} \tan^{-1} \frac{x+1}{2} + K$
(B) $\tan^{-1} (x+1) + K$
(C) $\frac{1}{2} \tan^{-1} (x+1) + K$
(D) $\log |x^2+2x+5| + K$
Topic 5: Integration by Parts and $e^x[f(x) + f'(x)]$ Form
$\int e^x (\sin x + \cos x) dx$ is equal to:
(A) $e^x \sin x + K$
(B) $e^x \cos x + K$
(C) $-e^x \sin x + K$
(D) $-e^x \cos x + K$
The value of $\int e^x (x^2 + 2x) dx$ is:
(A) $e^x \cdot x^2 + K$
(B) $e^x \cdot 2x + K$
(C) $\frac{e^x}{x^2} + K$
(D) $e^x + x^2 + K$
$\int e^x (\cos x – \sin x) dx$ is equal to:
(A) $e^x \cos x + K$
(B) $e^x \sin x + K$
(C) $-e^x \sin x + K$
(D) $-e^x \cos x + K$
The value of $\int e^x (\frac{1}{x} – \frac{1}{x^2}) dx$ is:
(A) $\frac{e^x}{x} + K$
(B) $-\frac{e^x}{x^2} + K$
(C) $e^x \log x + K$
(D) $\frac{e^x}{x^2} + K$
$\int e^x (\tan x + \log \sec x) dx$ is equal to:
(A) $e^x \log \sec x + K$
(B) $e^x \tan x + K$
(C) $e^x \sec x + K$
(D) $e^x + K$
The value of $\int e^x (\sec x + \sec x \tan x) dx$ is:
(A) $e^x \sec x + K$
(B) $e^x \tan x + K$
(C) $e^x \sec^2 x + K$
(D) $e^x + K$
$\int e^x (\tan^{-1} x + \frac{1}{1+x^2}) dx$ is equal to:
(A) $e^x \tan^{-1} x + K$
(B) $e^x \cot^{-1} x + K$
(C) $\frac{e^x}{1+x^2} + K$
(D) $e^x + K$
The formula for $\int e^x (f(x) + f'(x)) dx$ is:
(A) $e^x f(x) + K$
(B) $e^x f'(x) + K$
(C) $e^x + K$
(D) $f(x) + K$
$\int e^x (\cot x – \csc^2 x) dx$ is equal to:
(A) $e^x \cot x + K$
(B) $-e^x \csc^2 x + K$
(C) $e^x \csc x + K$
(D) $e^x + K$
The value of $\int e^x (\sin^{-1} x + \frac{1}{\sqrt{1-x^2}}) dx$ is:
(A) $e^x \sin^{-1} x + K$
(B) $e^x \cos^{-1} x + K$
(C) $\frac{e^x}{\sqrt{1-x^2}} + K$
(D) $e^x + K$
$\int x e^x dx$ is equal to:
(A) $e^x(x-1) + K$
(B) $e^x(x+1) + K$
(C) $x e^x + K$
(D) $e^x + K$
The value of $\int \log x dx$ is:
(A) $x \log x – x + K$
(B) $x \log x + x + K$
(C) $\frac{1}{x} + K$
(D) $\log x + K$
$\int x \sin x dx$ is equal to:
(A) $-x \cos x + \sin x + K$
(B) $x \cos x + \sin x + K$
(C) $-x \cos x – \sin x + K$
(D) $\sin x + K$
The value of $\int x \cos x dx$ is:
(A) $x \sin x + \cos x + K$
(B) $x \sin x – \cos x + K$
(C) $-x \sin x + \cos x + K$
(D) $\cos x + K$
$\int \sin^{-1} x dx$ is equal to:
(A) $x \sin^{-1} x + \sqrt{1-x^2} + K$
(B) $x \sin^{-1} x – \sqrt{1-x^2} + K$
(C) $\frac{1}{\sqrt{1-x^2}} + K$
(D) $\sin^{-1} x + K$
The value of $\int e^x (\frac{x-1}{x^2}) dx$ is:
(A) $\frac{e^x}{x} + K$
(B) $-\frac{e^x}{x} + K$
(C) $e^x \log x + K$
(D) $e^x x + K$
$\int \frac{x e^x}{(x+1)^2} dx$ is equal to:
(A) $\frac{e^x}{x+1} + K$
(B) $e^x(x+1) + K$
(C) $\frac{-e^x}{x+1} + K$
(D) $e^x + K$
The value of $\int e^x \frac{1+x \log x}{x} dx$ is:
(A) $e^x \log x + K$
(B) $\frac{e^x}{x} + K$
(C) $e^x(1+x) + K$
(D) $e^x + K$
$\int e^x (\sec x + \sec x \tan x) dx$ is equal to:
(A) $e^x \sec x + K$
(B) $e^x \tan x + K$
(C) $e^x \sec^2 x + K$
(D) $e^x + K$
The value of $\int e^x (\frac{1}{x^2} – \frac{2}{x^3}) dx$ is:
(A) $\frac{e^x}{x^2} + K$
(B) $-\frac{e^x}{x^3} + K$
(C) $e^x \log x + K$
(D) $e^x x^2 + K$
Topic 6: Basic Definite Integrals
The value of $\int_0^{\pi/2} \sin x dx$ is:
(A) 1
(B) 0
(C) -1
(D) $\pi/2$
$\int_0^{\pi/2} \cos x dx$ is equal to:
(A) 1
(B) 0
(C) -1
(D) $\pi/2$
The value of $\int_0^1 e^x dx$ is:
(A) $e-1$
(B) $e+1$
(C) $e$
(D) 1
$\int_1^2 x^2 dx$ is equal to:
(A) 7/3
(B) 8/3
(C) 1/3
(D) 3
The value of $\int_0^{\pi/4} \sec^2 x dx$ is:
(A) 1
(B) 0
(C) $\pi/4$
(D) -1
$\int_2^4 \frac{1}{x} dx$ is equal to:
(A) $\log 2$
(B) $\log 4$
(C) $2 \log 2$
(D) $\log(1/2)$
What is the value of $\int_0^1 \frac{dx}{1+x^2}$?
(A) $\pi/4$
(B) $\pi/2$
(C) 0
(D) 1
$\int_0^a x dx$ is equal to:
(A) $a^2/2$
(B) $a/2$
(C) $a^2$
(D) 0
The value of $\int_0^{\pi/2} \sin^2 x dx$ is:
(A) $\pi/4$
(B) $\pi/2$
(C) 1
(D) 0
$\int_0^{\pi/2} \cos^2 x dx$ is equal to:
(A) $\pi/4$
(B) $\pi/2$
(C) 1
(D) 0
The value of $\int_1^e \log x dx$ is:
(A) 1
(B) $e$
(C) $e-1$
(D) 0
$\int_0^{\pi} \cos x dx$ is equal to:
(A) 0
(B) 2
(C) 1
(D) -1
The value of $\int_0^1 \frac{dx}{\sqrt{1-x^2}}$ is:
(A) $\pi/2$
(B) $\pi/4$
(C) 1
(D) 0
$\int_2^3 x^3 dx$ is equal to:
(A) 65/4
(B) 81/4
(C) 16/4
(D) 50
The value of $\int_0^{\pi/2} \sin x \cos x dx$ is:
(A) 1/2
(B) 1
(C) 0
(D) 2
$\int_0^{\pi/4} \tan x dx$ is equal to:
(A) $\frac{1}{2} \log 2$
(B) $\log 2$
(C) 1
(D) 0
The value of $\int_0^1 (3x^2+2x+1) dx$ is:
(A) 3
(B) 6
(C) 1
(D) 0
What will be the value of $\int_{-1}^1 x dx$?
(A) 0
(B) 1
(C) 2
(D) -1
$\int_0^5 (x+1) dx$ is equal to:
(A) 17.5
(B) 15
(C) 20
(D) 10
The value of $\int_1^2 e^{2x} dx$ is:
(A) $\frac{e^4-e^2}{2}$
(B) $e^4-e^2$
(C) $\frac{e^2-e}{2}$
(D) 0
$\int_0^{\pi} \sin x dx$ is equal to:
(A) 2
(B) 0
(C) 1
(D) -2
The value of $\int_1^e \frac{1}{x} dx$ is:
(A) 1
(B) $e$
(C) 0
(D) $\log e$
What is the value of $\int_{-1}^1 x^3 dx$?
(A) 0
(B) 1/4
(C) 2
(D) -1
$\int_0^2 e^{x/2} dx$ is equal to:
(A) $2(e-1)$
(B) $e-1$
(C) $2e$
(D) 0
The value of $\int_0^{\pi/4} \sec x \tan x dx$ is:
(A) $\sqrt{2}-1$
(B) $\sqrt{2}$
(C) 1
(D) 0
$\int_0^1 (1-x)^9 dx$ is equal to:
(A) 1/10
(B) 1/9
(C) 1
(D) 0
The value of $\int_0^{\pi/2} \frac{\cos x}{1+\sin^2 x} dx$ is:
(A) $\pi/4$
(B) $\pi/2$
(C) 1
(D) 0
$\int_0^{\pi/2} \sin^2 \theta d\theta$ is equal to:
(A) $\pi/4$
(B) $\pi/2$
(C) 1
(D) 0
The value of $\int_2^3 \frac{1}{x} dx$ is:
(A) $\log(3/2)$
(B) $\log 3 – \log 2$
(C) Both (A) and (B)
(D) 0
$\int_0^1 \frac{e^x}{1+e^{2x}} dx$ is equal to:
(A) $\tan^{-1} e – \pi/4$
(B) $\tan^{-1} e$
(C) $\pi/4$
(D) 0
The value of $\int_1^2 \frac{dx}{x^2}$ is:
(A) 1/2
(B) -1/2
(C) 1
(D) 0
$\int_0^{\pi/2} \sin x \cos^2 x dx$ is equal to:
(A) 1/3
(B) 1/2
(C) 1
(D) 0
The value of $\int_0^{\pi/2} \frac{dx}{1+\cos x}$ is:
(A) 1
(B) 2
(C) $\pi/2$
(D) 0
$\int_0^1 x(1-x)^5 dx$ is equal to:
(A) 1/42
(B) 1/30
(C) 1/6
(D) 0
The value of $\int_0^{\pi/2} \cos^3 x dx$ is:
(A) 2/3
(B) 1/3
(C) $\pi/4$
(D) 0
$\int_1^4 \sqrt{x} dx$ is equal to:
(A) 14/3
(B) 7/3
(C) 4/3
(D) 0
The value of $\int_0^1 \frac{dx}{x+1}$ is:
(A) $\log 2$
(B) 1
(C) 0
(D) $\log 1$
$\int_0^1 e^{-2x} dx$ is equal to:
(A) $\frac{1-e^{-2}}{2}$
(B) $1-e^{-2}$
(C) $e^{-2}$
(D) 0
The value of $\int_0^{\pi/2} \frac{d\theta}{1+\sin \theta}$ is:
(A) 1
(B) 2
(C) $\pi/2$
(D) 0
$\int_0^1 \frac{dx}{\sqrt{1+x}+\sqrt{x}}$ is equal to:
(A) $\frac{2}{3}(2\sqrt{2}-1)$
(B) $\frac{2}{3}$
(C) $\sqrt{2}-1$
(D) 0
Topic 7: Properties of Definite Integrals
$\int_{-a}^a f(x) dx = 0$ if $f(x)$ is an:
(A) Odd function
(B) Even function
(C) Periodic function
(D) None of these
The value of $\int_{-1}^1 \sin^5 x \cos^4 x dx$ is:
(A) 0
(B) 1
(C) 2
(D) $\pi/2$
$\int_{-\pi/2}^{\pi/2} \sin^7 x dx$ is equal to:
(A) 0
(B) 1
(C) $\pi$
(D) -1
The value of $\int_0^{\pi/2} \frac{\sin x}{\sin x + \cos x} dx$ is:
(A) $\pi/4$
(B) $\pi/2$
(C) 0
(D) 1
$\int_0^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx$ is equal to:
(A) $\pi/4$
(B) $\pi/2$
(C) 0
(D) $\pi$
What will be the value of $\int_0^{\pi/2} \frac{dx}{1+\tan x}$?
(A) $\pi/4$
(B) $\pi/2$
(C) 0
(D) 1
$\int_0^{\pi} \frac{x \sin x}{1+\cos^2 x} dx$ is equal to:
(A) $\pi^2/4$
(B) $\pi^2/2$
(C) $\pi/4$
(D) 0
The value of $\int_{-1}^1 |x| dx$ is:
(A) 1
(B) 0
(C) 2
(D) 1/2
$\int_0^2 |x-1| dx$ is equal to:
(A) 1
(B) 2
(C) 0
(D) 0.5
The value of $\int_0^{\pi} \sin^2 x dx$ is:
(A) $\pi/2$
(B) $\pi$
(C) $\pi/4$
(D) 0
$\int_0^{\pi} \cos^3 x dx$ is equal to:
(A) 0
(B) 1
(C) -1
(D) $\pi$
$\int_0^a f(x) dx$ is equal to?
(A) $\int_0^a f(a-x) dx$
(B) $\int_0^a f(x-a) dx$
(C) $-\int_0^a f(x) dx$
(D) 0
The value of $\int_{-\pi/4}^{\pi/4} \tan x dx$ is:
(A) 0
(B) 1
(C) $\log 2$
(D) $\pi/4$
$\int_0^{\pi/2} \log \sin x dx$ is equal to:
(A) $-\frac{\pi}{2} \log 2$
(B) $\frac{\pi}{2} \log 2$
(C) 0
(D) $\pi \log 2$
The value of $\int_0^{\pi/2} \log \cos x dx$ is:
(A) $-\frac{\pi}{2} \log 2$
(B) 0
(C) $\pi/2$
(D) 1
$\int_0^1 x(1-x)^n dx$ is equal to:
(A) $\frac{1}{(n+1)(n+2)}$
(B) $\frac{1}{n+1}$
(C) $\frac{1}{n+2}$
(D) 0
The value of $\int_0^{\pi/2} \frac{\cos x – \sin x}{1 + \sin x \cos x} dx$ is:
(A) 0
(B) 1
(C) $\pi/2$
(D) $\pi/4$
$\int_0^{\pi/2} \frac{\sin^n x}{\sin^n x + \cos^n x} dx$ is equal to:
(A) $\pi/4$
(B) $\pi/2$
(C) 0
(D) 1
$\int_{-a}^a x^n dx$ if $n$ is odd:
(A) 0
(B) $\frac{2a^{n+1}}{n+1}$
(C) $a^n$
(D) 1
$\int_0^{2a} f(x) dx = 2\int_0^a f(x) dx$ if:
(A) $f(2a-x) = f(x)$
(B) $f(2a-x) = -f(x)$
(C) $f(x)$ is odd
(D) None of these
The value of $\int_{-\pi/2}^{\pi/2} \sin^2 x dx$ is:
(A) $\pi/2$
(B) $\pi$
(C) 0
(D) 1
$\int_{-1}^1 x |x| dx$ is equal to:
(A) 0
(B) 2/3
(C) 1
(D) -1
What is the value of $\int_0^a \frac{f(x)}{f(x)+f(a-x)} dx$?
(A) $a/2$
(B) $a$
(C) 0
(D) $2a$
$\int_0^{\pi} \frac{dx}{1+2\sin^2 x}$ is equal to:
(A) $\pi/\sqrt{3}$
(B) $\pi/3$
(C) $\pi/2$
(D) 0
$\int_{-a}^a f(x) dx = 2\int_0^a f(x) dx$ if $f(x)$ is:
(A) Even function
(B) Odd function
(C) Zero
(D) Constant
The value of $\int_0^{\pi} \cos^5 x dx$ is:
(A) 0
(B) 1
(C) 2
(D) -1
$\int_{-\pi/2}^{\pi/2} \sin^3 x dx$ is equal to:
(A) 0
(B) 1
(C) 2
(D) -1
What will be the value of $\int_0^2 |x-1| dx$?
(A) 1
(B) 2
(C) 0
(D) 0.5
The value of $\int_0^{\pi/2} \log \tan x dx$ is:
(A) 0
(B) $\pi/2$
(C) $\pi/4$
(D) 1
$\int_{-1}^1 x^{17} \cos^4 x dx$ is equal to:
(A) 0
(B) 1
(C) $\pi/2$
(D) 2
Topic 8: Mixed Advanced Integrals
The value of $\int \frac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} dx$ is:
(A) $x + K$
(B) $\sin x – \cos x + K$
(C) $\log |x| + K$
(D) 0
$\int \frac{dx}{x(x^n+1)}$ is equal to:
(A) $\frac{1}{n} \log |\frac{x^n}{x^n+1}| + K$
(B) $\log |\frac{x^n}{x^n+1}| + K$
(C) $n \log |x^n+1| + K$
(D) 0
The value of $\int \frac{dx}{\cos^2 x (1-\tan x)^2}$ is:
(A) $\frac{1}{1-\tan x} + K$
(B) $\frac{-1}{1-\tan x} + K$
(C) $\log |1-\tan x| + K$
(D) $\tan x + K$
$\int \frac{dx}{e^x + e^{-x}}$ is equal to:
(A) $\tan^{-1} (e^x) + K$
(B) $\log |e^x + e^{-x}| + K$
(C) $e^x + K$
(D) $\tan^{-1} (e^{-x}) + K$
The value of $\int \frac{\cos \sqrt{x}}{\sqrt{x}} dx$ is:
(A) $2 \sin \sqrt{x} + K$
(B) $\sin \sqrt{x} + K$
(C) $\frac{1}{2} \sin \sqrt{x} + K$
(D) $2 \cos \sqrt{x} + K$
$\int \frac{x^3}{x+1} dx$ is equal to:
(A) $\frac{x^3}{3} – \frac{x^2}{2} + x – \log |x+1| + K$
(B) $\frac{x^4}{4} + K$
(C) $\log |x+1| + K$
(D) $x^2 + K$
The value of $\int \frac{\sin 2x}{a \cos^2 x + b \sin^2 x} dx$ is:
(A) $\frac{1}{b-a} \log |a \cos^2 x + b \sin^2 x| + K$
(B) $\log |a \cos^2 x + b \sin^2 x| + K$
(C) $\frac{1}{a-b} \tan x + K$
(D) 0
$\int \frac{dx}{\sqrt{1+x}-\sqrt{x}}$ is equal to:
(A) $\frac{2}{3} [(1+x)^{3/2} + x^{3/2}] + K$
(B) $\frac{2}{3} [(1+x)^{3/2} – x^{3/2}] + K$
(C) $\sqrt{1+x} + \sqrt{x} + K$
(D) 0
The formula for $\int \sqrt{a^2-x^2} dx$ is:
(A) $\frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + K$
(B) $\frac{x}{2}\sqrt{a^2-x^2} – \frac{a^2}{2}\sin^{-1}\frac{x}{a} + K$
(C) $a^2 \sin^{-1} \frac{x}{a} + K$
(D) 0
$\int \sqrt{x^2+a^2} dx$ is equal to:
(A) $\frac{x}{2}\sqrt{x^2+a^2} + \frac{a^2}{2}\log|x+\sqrt{x^2+a^2}| + K$
(B) $\frac{x}{2}\sqrt{x^2+a^2} – \frac{a^2}{2}\log|x+\sqrt{x^2+a^2}| + K$
(C) $\log|x+\sqrt{x^2+a^2}| + K$
(D) 0
The value of $\int \sqrt{x^2-a^2} dx$ is:
(A) $\frac{x}{2}\sqrt{x^2-a^2} – \frac{a^2}{2}\log|x+\sqrt{x^2-a^2}| + K$
(B) $\frac{x}{2}\sqrt{x^2-a^2} + \frac{a^2}{2}\log|x+\sqrt{x^2-a^2}| + K$
(C) $\log|x+\sqrt{x^2-a^2}| + K$
(D) 0
$\int \frac{\sin 2x}{\sin^4 x + \cos^4 x} dx$ is equal to:
(A) $\tan^{-1} (\tan^2 x) + K$
(B) $\tan^{-1} (\sin^2 x) + K$
(C) $\log |\sin^4 x + \cos^4 x| + K$
(D) 0
The value of $\int \tan^{-1} \sqrt{x} dx$ is:
(A) $(x+1) \tan^{-1} \sqrt{x} – \sqrt{x} + K$
(B) $x \tan^{-1} \sqrt{x} + K$
(C) $\frac{1}{1+x} + K$
(D) 0
$\int \frac{dx}{\sqrt{x+a}+\sqrt{x+b}}$ is equal to:
(A) $\frac{2}{3(a-b)} [(x+a)^{3/2} – (x+b)^{3/2}] + K$
(B) $\frac{2}{3} [(x+a)^{3/2} + (x+b)^{3/2}] + K$
(C) $\sqrt{x+a} – \sqrt{x+b} + K$
(D) 0
$\int \frac{dx}{\sin(x-a)\sin(x-b)}$ is equal to:
(A) $\frac{1}{\sin(a-b)} \log |\frac{\sin(x-a)}{\sin(x-b)}| + K$
(B) $\log |\frac{\sin(x-a)}{\sin(x-b)}| + K$
(C) $\sin(a-b) \tan(x-a) + K$
(D) 0
$\int \frac{\sin^6 x + \cos^6 x}{\sin^2 x \cos^2 x} dx$ is equal to:
(A) $\tan x – \cot x – 3x + K$
(B) $\tan x + \cot x + 3x + K$
(C) $\sin x + \cos x + K$
(D) 0
The value of $\int \frac{x^2+1}{x^4+1} dx$ is:
(A) $\frac{1}{\sqrt{2}} \tan^{-1} (\frac{x^2-1}{\sqrt{2}x}) + K$
(B) $\tan^{-1} x^2 + K$
(C) $\log |x^4+1| + K$
(D) 0
$\int \frac{dx}{x(x^5+1)}$ is equal to:
(A) $\frac{1}{5} \log |\frac{x^5}{x^5+1}| + K$
(B) $\log |\frac{x^5}{x^5+1}| + K$
(C) $5 \log |x^5+1| + K$
(D) 0
The value of $\int \sqrt{\tan x} dx$ is:
(A) $\frac{1}{\sqrt{2}} \tan^{-1} (\frac{\tan x – 1}{\sqrt{2 \tan x}}) + \frac{1}{2\sqrt{2}} \log |\frac{\tan x – \sqrt{2 \tan x} + 1}{\tan x + \sqrt{2 \tan x} + 1}| + K$
(B) $2\sqrt{\tan x} + K$
(C) $\sec^2 x + K$
(D) 0
$\int \frac{dx}{\sin x + \sqrt{3} \cos x}$ is equal to:
(A) $\frac{1}{2} \log |\tan(\frac{x}{2} + \frac{\pi}{6})| + K$
(B) $\log |\sin x + \cos x| + K$
(C) $\frac{\pi}{6} + K$
(D) 0
The value of $\int \frac{dx}{5+4 \cos x}$ is:
(A) $\frac{2}{3} \tan^{-1} (\frac{1}{3} \tan \frac{x}{2}) + K$
(B) $\tan^{-1} (\tan \frac{x}{2}) + K$
(C) $\frac{1}{3} \log |5+4 \cos x| + K$
(D) 0
$\int \frac{2 \sin x + 3 \cos x}{3 \sin x + 4 \cos x} dx$ is equal to:
(A) $\frac{18}{25}x + \frac{1}{25} \log |3 \sin x + 4 \cos x| + K$
(B) $x + \log |3 \sin x + 4 \cos x| + K$
(C) $\tan x + K$
(D) 0
The value of $\int \frac{dx}{(x+1)\sqrt{x+2}}$ is:
(A) $\log |\frac{\sqrt{x+2}-1}{\sqrt{x+2}+1}| + K$
(B) $2 \tan^{-1} \sqrt{x+2} + K$
(C) $\sqrt{x+2} + K$
(D) 0
$\int \frac{dx}{\sqrt{(x-a)(b-x)}}$ is equal to:
(A) $2 \sin^{-1} \sqrt{\frac{x-a}{b-a}} + K$
(B) $\sin^{-1} \frac{x-a}{b-a} + K$
(C) $\log |(x-a)(b-x)| + K$
(D) 0
The value of $\int_0^{\pi/2} \sin^{10} x dx$ is:
(A) $\frac{9 \cdot 7 \cdot 5 \cdot 3 \cdot 1}{10 \cdot 8 \cdot 6 \cdot 4 \cdot 2} \cdot \frac{\pi}{2}$
(B) $\frac{9}{10} \cdot \frac{\pi}{2}$
(C) 1
(D) 0
$\int_0^{\pi/2} \cos^7 x dx$ is equal to:
(A) $\frac{6 \cdot 4 \cdot 2}{7 \cdot 5 \cdot 3 \cdot 1}$
(B) $\frac{6}{7} \cdot \frac{\pi}{2}$
(C) 1
(D) 0
$\int_0^{\pi} \sin mx \sin nx dx$ where $m \ne n$:
(A) 0
(B) $\pi/2$
(C) $\pi$
(D) 1
The value of $\int_0^{\pi/2} \frac{dx}{1+\tan^3 x}$ is:
(A) $\pi/4$
(B) $\pi/2$
(C) 0
(D) 1
$\int_0^1 \frac{\log(1+x)}{1+x^2} dx$ is equal to:
(A) $\frac{\pi}{8} \log 2$
(B) $\frac{\pi}{4} \log 2$
(C) $\log 2$
(D) 0
The value of $\int_0^{\pi} x \sin^3 x dx$ is:
(A) $2\pi/3$
(B) $\pi/3$
(C) $4\pi/3$
(D) 0
$\int_0^{\pi/2} \frac{\sin^2 x}{\sin x + \cos x} dx$ is equal to:
(A) $\frac{1}{\sqrt{2}} \log(\sqrt{2}+1)$
(B) $\sqrt{2} \log(\sqrt{2}+1)$
(C) $\pi/4$
(D) 0
The value of $\int_0^{\pi} \frac{x dx}{a^2 \cos^2 x + b^2 \sin^2 x}$ is:
(A) $\frac{\pi^2}{2ab}$
(B) $\frac{\pi}{2ab}$
(C) $\frac{\pi^2}{ab}$
(D) 0
$\int_0^{\infty} \frac{dx}{1+x^2}$ is equal to:
(A) $\pi/2$
(B) $\pi$
(C) $\infty$
(D) 0
What will be the value of $\int_0^1 x e^x dx$?
(A) 1
(B) $e$
(C) $e-1$
(D) 0
The value of $\int_1^2 \log x dx$ is:
(A) $2 \log 2 – 1$
(B) $\log 2$
(C) $2 \log 2 + 1$
(D) 0
$\int_0^{\pi/2} \sin^3 x \cos x dx$ is equal to:
(A) 1/4
(B) 1/2
(C) 1
(D) 0
The value of $\int_0^{\pi/4} \tan^2 x dx$ is:
(A) $1 – \pi/4$
(B) $1 + \pi/4$
(C) $\pi/4$
(D) 0
$\int_0^{\pi/2} e^x (\sin x + \cos x) dx$ is equal to:
(A) $e^{\pi/2}$
(B) $e^{\pi/2}-1$
(C) 1
(D) 0
The value of $\int_0^1 \frac{x dx}{x^2+1}$ is:
(A) $\frac{1}{2} \log 2$
(B) $\log 2$
(C) $\pi/4$
(D) 0
$\int \sec^2 (7-4x) dx$ is equal to:
(A) $-\frac{1}{4} \tan (7-4x) + K$
(B) $\frac{1}{4} \tan (7-4x) + K$
(C) $\tan (7-4x) + K$
(D) $-4 \tan (7-4x) + K$
The value of $\int \frac{1}{1-\sin x} dx$ is:
(A) $\tan x + \sec x + K$
(B) $\tan x – \sec x + K$
(C) $\sec x – \tan x + K$
(D) $\tan x + K$
$\int \sqrt{1+\cos 2x} dx$ is equal to:
(A) $\sqrt{2} \sin x + K$
(B) $\sqrt{2} \cos x + K$
(C) $2 \sin x + K$
(D) $\sin x + K$
The value of $\int \frac{\cos 2x}{(\sin x + \cos x)^2} \, dx$ is:
(A) $\log(\sin x + \cos x) + K$
(B) $2 \log(\sin x + \cos x) + K$
(C) $\log(\sin x – \cos x) + K$
(D) $-\frac{1}{\sin x + \cos x} + K$
$\int \frac{dx}{x^2+2x}$ is equal to:
(A) $\frac{1}{2} \log |\frac{x}{x+2}| + K$
(B) $\log |\frac{x}{x+2}| + K$
(C) $\frac{1}{2} \tan^{-1} (x+1) + K$
(D) 0
The value of $\int \frac{\sin x}{\sin(x-\alpha)} dx$ is:
(A) $x \cos \alpha + \sin \alpha \log |\sin(x-\alpha)| + K$
(B) $x \sin \alpha + \cos \alpha \log |\sin(x-\alpha)| + K$
(C) $\log |\sin(x-\alpha)| + K$
(D) 0
$\int \frac{dx}{1+\sin x}$ is equal to:
(A) $\tan x – \sec x + K$
(B) $\tan x + \sec x + K$
(C) $\sec x – \tan x + K$
(D) 0
The value of $\int \frac{\cos 2x – 1}{\cos 2x + 1} dx$ is:
(A) $x – \tan x + K$
(B) $\tan x – x + K$
(C) $x + \tan x + K$
(D) 0
$\int \frac{dx}{\sqrt{1+\sin x}}$ is equal to:
(A) $\sqrt{2} \log |\tan(\frac{x}{4} + \frac{\pi}{8})| + K$
(B) $2\sqrt{1+\sin x} + K$
(C) $\cos x + K$
(D) 0
What will be the value of $\int \frac{x^2}{x^2+1} dx$?
(A) $x – \tan^{-1} x + K$
(B) $x + \tan^{-1} x + K$
(C) $\tan^{-1} x + K$
(D) 0
$\int \frac{dx}{x^2-1}$ is equal to:
(A) $\frac{1}{2} \log |\frac{x-1}{x+1}| + K$
(B) $\frac{1}{2} \log |\frac{x+1}{x-1}| + K$
(C) $\log |x^2-1| + K$
(D) 0
The value of $\int \frac{dx}{1-x^2}$ is:
(A) $\frac{1}{2} \log |\frac{1+x}{1-x}| + K$
(B) $\frac{1}{2} \log |\frac{1-x}{1+x}| + K$
(C) $\tan^{-1} x + K$
(D) 0
$\int e^{2x} \sin x dx$ is equal to:
(A) $\frac{e^{2x}}{5} (2\sin x – \cos x) + K$
(B) $\frac{e^{2x}}{5} (2\sin x + \cos x) + K$
(C) $e^{2x} \sin x + K$
(D) 0
The value of $\int \frac{dx}{x \sqrt{1+x^n}}$ is:
(A) $\frac{2}{n} \log |\frac{\sqrt{1+x^n}-1}{\sqrt{1+x^n}+1}| + K$
(B) $\log |\sqrt{1+x^n}| + K$
(C) $\frac{1}{n} \tan^{-1} \sqrt{x^n} + K$
(D) 0
$\int \frac{dx}{x^4-1}$ is equal to:
(A) $\frac{1}{4} \log |\frac{x-1}{x+1}| – \frac{1}{2} \tan^{-1} x + K$
(B) $\frac{1}{4} \log |\frac{x-1}{x+1}| + \frac{1}{2} \tan^{-1} x + K$
(C) $\log |x^4-1| + K$
(D) 0
The value of $\int \frac{dx}{x(1+x^2)}$ is:
(A) $\log |x| – \frac{1}{2} \log (1+x^2) + K$
(B) $\log |x| + \log (1+x^2) + K$
(C) $\tan^{-1} x + K$
(D) 0
$\int \frac{dx}{e^x-1}$ is equal to:
(A) $\log |e^x-1| – x + K$
(B) $\log |e^x-1| + K$
(C) $x – \log |e^x-1| + K$
(D) 0
The value of $\int \frac{dx}{x^2+4x+13}$ is:
(A) $\frac{1}{3} \tan^{-1} \frac{x+2}{3} + K$
(B) $\tan^{-1} \frac{x+2}{3} + K$
(C) $\frac{1}{3} \log |x^2+4x+13| + K$
(D) 0
$\int \frac{x dx}{(x-1)(x-2)}$ is equal to:
(A) $2 \log |x-2| – \log |x-1| + K$
(B) $\log |x-2| – 2 \log |x-1| + K$
(C) $\log |(x-1)(x-2)| + K$
(D) 0
What is the value of $\int \frac{dx}{x^2-2x+2}$?
(A) $\tan^{-1} (x-1) + K$
(B) $\tan^{-1} x + K$
(C) $\log |x^2-2x+2| + K$
(D) 0
$\int \frac{dx}{\sqrt{1+x-x^2}}$ is equal to:
(A) $\sin^{-1} (\frac{2x-1}{\sqrt{5}}) + K$
(B) $\sin^{-1} (\frac{x-1}{\sqrt{5}}) + K$
(C) $\log |x + \sqrt{1+x-x^2}| + K$
(D) 0
The value of $\int \frac{dx}{x \sqrt{x^4-1}}$ is:
(A) $\frac{1}{2} \sec^{-1} (x^2) + K$
(B) $\sec^{-1} (x^2) + K$
(C) $\frac{1}{2} \tan^{-1} (x^2) + K$
(D) 0
$\int \frac{1+\cos x}{1-\cos x} dx$ is equal to:
(A) $-2 \cot(x/2) – x + K$
(B) $2 \tan(x/2) – x + K$
(C) $-2 \csc x + K$
(D) 0
The value of $\int \frac{\sin x}{1+\cos^2 x} dx$ is:
(A) $-\tan^{-1} (\cos x) + K$
(B) $\tan^{-1} (\cos x) + K$
(C) $\log |1+\cos^2 x| + K$
(D) 0
$\int \frac{dx}{x \log x \log(\log x)}$ is equal to:
(A) $\log |\log (\log x)| + K$
(B) $\log (\log x) + K$
(C) $(\log x)^2 + K$
(D) 0
The value of $\int \frac{dx}{\sqrt{x^2+2x+2}}$ is:
(A) $\log |x+1 + \sqrt{x^2+2x+2}| + K$
(B) $\sin^{-1} (x+1) + K$
(C) $\log |x+\sqrt{x^2+2x+2}| + K$
(D) 0
$\int \frac{e^x(1+x)}{\cos^2(x e^x)} dx$ is equal to:
(A) $\tan(x e^x) + K$
(B) $\cot(x e^x) + K$
(C) $\sec(x e^x) + K$
(D) 0
What is the value of $\int \frac{dx}{\sin(x-a)\cos(x-a)}$?
(A) $\log |\tan(x-a)| + K$
(B) $\log |\sin(x-a)| + K$
(C) $\log |\cos(x-a)| + K$
(D) 0
Bihar Board Class 12th के (Mathematics/गणित) = गणित ‘भाग-2 (Englsih Medium) Book Chapter- 7 Integration के Exam 2027 MCQs Questions Answer Key
| Q.No | Ans | Q.No | Ans | Q.No | Ans | Q.No | Ans |
| 1 | A | 63 | A | 125 | A | 187 | A |
| 2 | A | 64 | A | 126 | A | 188 | A |
| 3 | A | 65 | A | 127 | A | 189 | A |
| 4 | A | 66 | A | 128 | A | 190 | A |
| 5 | A | 67 | A | 129 | A | 191 | A |
| 6 | A | 68 | A | 130 | A | 192 | A |
| 7 | A | 69 | A | 131 | A | 193 | A |
| 8 | A | 70 | A | 132 | A | 194 | A |
| 9 | A | 71 | A | 133 | A | 195 | A |
| 10 | A | 72 | A | 134 | A | 196 | A |
| 11 | A | 73 | A | 135 | A | 197 | A |
| 12 | A | 74 | A | 136 | A | 198 | A |
| 13 | A | 75 | A | 137 | A | 199 | A |
| 14 | A | 76 | A | 138 | A | 200 | A |
| 15 | A | 77 | A | 139 | C | 201 | A |
| 16 | A | 78 | A | 140 | A | 202 | A |
| 17 | A | 79 | A | 141 | A | 203 | A |
| 18 | A | 80 | A | 142 | A | 204 | A |
| 19 | A | 81 | A | 143 | A | 205 | A |
| 20 | A | 82 | A | 144 | A | 206 | A |
| 21 | A | 83 | A | 145 | A | 207 | A |
| 22 | A | 84 | A | 146 | A | 208 | A |
| 23 | A | 85 | A | 147 | A | 209 | A |
| 24 | A | 86 | A | 148 | A | 210 | A |
| 25 | A | 87 | A | 149 | A | 211 | A |
| 26 | A | 88 | A | 150 | A | 212 | A |
| 27 | A | 89 | A | 151 | A | 213 | A |
| 28 | A | 90 | A | 152 | A | 214 | A |
| 29 | A | 91 | A | 153 | A | 215 | A |
| 30 | A | 92 | A | 154 | A | 216 | A |
| 31 | A | 93 | A | 155 | A | 217 | A |
| 32 | A | 94 | A | 156 | A | 218 | B |
| 33 | A | 95 | A | 157 | A | 219 | A |
| 34 | A | 96 | A | 158 | A | 220 | A |
| 35 | A | 97 | A | 159 | A | 221 | A |
| 36 | A | 98 | A | 160 | A | 222 | A |
| 37 | A | 99 | A | 161 | A | 223 | A |
| 38 | A | 100 | A | 162 | A | 224 | A |
| 39 | A | 101 | A | 163 | A | 225 | A |
| 40 | A | 102 | A | 164 | A | 226 | C |
| 41 | A | 103 | A | 165 | A | 227 | B |
| 42 | B | 104 | A | 166 | A | 228 | A |
| 43 | A | 105 | A | 167 | A | 229 | A |
| 44 | A | 106 | A | 168 | A | 230 | A |
| 45 | A | 107 | A | 169 | A | 231 | A |
| 46 | A | 108 | A | 170 | A | 232 | A |
| 47 | A | 109 | A | 171 | A | 233 | A |
| 48 | A | 110 | A | 172 | A | 234 | A |
| 49 | A | 111 | A | 173 | A | 235 | A |
| 50 | A | 112 | A | 174 | A | 236 | A |
| 51 | A | 113 | A | 175 | A | 237 | A |
| 52 | A | 114 | A | 176 | A | 238 | A |
| 53 | A | 115 | A | 177 | A | 239 | A |
| 54 | A | 116 | A | 178 | A | 240 | A |
| 55 | A | 117 | A | 179 | A | 241 | A |
| 56 | A | 118 | A | 180 | A | 242 | A |
| 57 | A | 119 | A | 181 | A | 243 | A |
| 58 | A | 120 | A | 182 | A | 244 | A |
| 59 | A | 121 | A | 183 | B | 245 | A |
| 60 | A | 122 | A | 184 | A | 246 | A |
| 61 | A | 123 | A | 185 | A | 247 | A |
| 62 | A | 124 | A | 186 | A |
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