# MCQ Questions for Class 12 Maths Chapter 7 Integrals with Answers

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## Integrals Class 12 MCQs Questions with Answers

Don’t forget to practice the multitude of MCQ Questions on Integrals Class 12 with answers so you can apply your skills during the exam.

Question 1.
The anti-derivative of (√x + $$\frac { 1 }{√x}$$) equals

Answer: (c) $$\frac { 2 }{3}$$ x$$\frac { 2 }{3}$$ + 2x$$\frac { 1 }{2}$$ + c

Question 2.
If $$\frac { 1 }{dx}$$ (f(x)) = 4x³ – $$\frac { 3 }{x^4}$$ such that f(2) = 0 then f(x) is ……………

Answer: (a) x4 + $$\frac { 1 }{x^3}$$ – $$\frac { 129 }{8}$$

Question 3.
∫$$\frac { 10x^9+10^x log_e 10 }{x^{10} + 10^x}$$ dx equals
(a) 10x -x10 + c
(b) 10x + x10 + c
(c) (10x – x10)-1 + c
(d) log (10x + x10) + c.

Answer: (d) log (10x + x10) + c.

Question 4.
∫$$\frac { dx }{sin^2 x cos^2 x}$$ equals
(a) tan x + cot x + c
(b) tan x – cot x + c
(c) tan x cot x + c
(d) tan x – cot 2x + c.

Answer: (b) tan x – cot x + c

Question 5.
∫$$\frac { sin^2 x – cos ^2 x }{sin^2 x cos^2 x}$$ dx is equals to
(a) tan x + cot x + c
(b) tan x + cosec x + c
(c) -tan x + cot x + c
(d) tan x + sec x + c.

Answer: (a) tan x + cot x + c

Question 6.
∫$$\frac { e^x(1 + x) }{cos^2(xe^2)}$$ dx is equals to
(a) -cot (xex) + c
(b) tan (xex) + c
(c) tan (ex) + c
(d) cot (ex) + c

Answer: (b) tan (xex) + c

Question 7.
∫$$\frac { dx }{x^2+2x+2}$$ equals
(a) x tan-1 (x + 1) + c
(b) tan-1 (x + 1) + c
(c) (x + 1) tan-1 x + c
(d) tan-1 x + c.

Answer: (b) tan-1 (x + 1) + c

Question 8.
∫$$\frac { dx }{\sqrt{9-25x^2}}$$ equals

Answer: (b) $$\frac { 1 }{5}$$ sin-1 ($$\frac { 5x }{3}$$) + c

Question 9.
∫$$\frac { x dx }{(x-1)(x-2)}$$ equals

(d) log |(x – 1) (x – 2)| + c.

Answer: (b) log |$$\frac { (x-2)^2 }{x-1}$$| + c

Question 10.
∫$$\frac { dx }{x(x^2+1)}$$ equals
(a) log |x| – $$\frac { 1 }{2}$$ log (x² + 1) + c
(b) $$\frac { 1 }{2}$$ log |x| + $$\frac { 1 }{2}$$ log (x² + 1) + c
(c) -log |x| + $$\frac { 1 }{2}$$ log (x² + 1) + c
(d) log |x| + log (x² + 1) + c

Answer: (a) log |x| – $$\frac { 1 }{2}$$ log (x² + 1) + c

Question 11.
∫x² e dx equals
(a) $$\frac { 1 }{3}$$ e + c
(b) $$\frac { 1 }{3}$$ e + c
(c) $$\frac { 1 }{2}$$ e + c
(d) $$\frac { 1 }{2}$$ e + c

Answer: (a) $$\frac { 1 }{3}$$ e + c

Question 12.
∫ex sec x (1 + tan x) dx equals
(a) ex cos x + c
(b) ex sec x + c
(c) ex sin x + c
(d) ex tan x + c.

Answer: (b) ex sec x + c

Question 13.
∫$$\sqrt { 1 + x^2}$$ dx is equal to

Question 14.
∫$$\sqrt { x^2 – 8x + 7}$$ dx is equal to

Question 15.
$$\int_{1}^{\sqrt{3}}$$ $$\frac { dx }{1+x^2}$$ equals
(a) $$\frac { π }{3}$$
(b) $$\frac { 2π }{3}$$
(c) $$\frac { π }{6}$$
(d) $$\frac { π }{112}$$

Answer: (d) $$\frac { π }{112}$$

Question 16.
$$\int_{1}^{2/3}$$ $$\frac { dx }{4+9x^2}$$ equals
(a) $$\frac { π }{6}$$
(b) $$\frac { π }{12}$$
(c) $$\frac { π }{24}$$
(d) $$\frac { π }{4}$$

Answer: (c) $$\frac { π }{24}$$

Question 17.
The value of the integral $$\int_{1}^{2/3}$$ $$\frac { (x-x^3)^{1/3} }{x^4}$$ dx is
(a) 6
(b) 0
(c) 3
(d) 4

Question 18.
If f(x) = $$\int_{0}^{x}$$ t sin t dt, then f'(x) is
(a) cos x + x sin x
(b) x sin x
(c) x cos x
(d) sin x + x cos x.

Question 19.
The value of
$$\int_{-π/2}^{π/2}$$ (x³ + x cos x + tan5 x + 1) dx is
(a) 0
(b) 2
(c) π
(d) 1

Question 20.
The value of $$\int_{0}^{π/2}$$ log ($$\frac { 4+3 sin x }{4+3 cos x}$$) dx is
(a) 2
(b) $$\frac { 3 }{4}$$
(c) 0
(d) -2

Question 21.
∫$$\frac { dx }{e^x+e{-x}}$$ is equal to
(a) tan-1 (ex) + c
(b) tan-1 (e-x) + c
(c) log (ex – e-1) + c
(d) log (ex + e-x) + c.

Answer: (a) tan-1 (ex) + c

Question 22.
∫$$\frac { cos 2x }{(sin x + cos x)^2}$$ dx is equal to
(a) $$\frac { -1 }{sin x + cos x}$$ + c
(b) log |sin x + cos x| + c
(c) log |sin x – cos x| + c
(d) $$\frac { 1 }{(sin x + cos x)^2}$$ + c

Answer: (b) log |sin x + cos x| + c

Question 23.
If f (a + b – x) = f(x), then $$\int_{a}^{b}$$ x f(x) dx is equal to

Answer: (d) $$\frac { a+b }{2}$$ $$\int_{a}^{b}$$ f(x) dx

Question 24.
∫ex(cos x – sin x)dx is equal to
(a) ex – cos x + c
(b) ex sin x + c
(c) -ex cos x + c
(d) -ex sin x + c.

Answer: (a) ex – cos x + c

Question 25.
∫$$\frac { dx }{sin^2 x cos^2 x}$$ is equal to
(a) tan x + cot x + c
(b) (tan x + cot x)² + c
(c) tan x – cot x + c
(d) (tan x – cot x)² + c.

Answer: (c) tan x – cot x + c

Question 26.
If ∫ $$\frac { 3e^x-5e^{-x} }{4r^x+5e^{-x}}$$ dx = ax + b log |4ex + 5e-x| + c then
(a) a = –$$\frac { 1 }{8}$$, b = $$\frac { 7 }{8}$$
(b) a = $$\frac { 1 }{8}$$, b = $$\frac { 7 }{8}$$
(c) a = $$\frac { -1 }{8}$$, b = –$$\frac { 7 }{8}$$
(d) a = $$\frac { 1 }{8}$$, b = –$$\frac { 7 }{8$$

Answer: (a) a = –$$\frac { 1 }{8}$$, b = $$\frac { 7 }{8}$$

Question 27.
∫tan-1 √x dx is equal to
(a) (x + 1)tan-1 √x – √x + c
(b) x tan-1 √x – √x + c
(c) √x – x tan-1 √x + c
(d) -1x – (x + 1) tan-1 √x + c

Answer: (a) (x + 1)tan-1 √x – √x + c

Question 28.
∫ex($$\frac { 1-x }{(1+x^2)}$$)2 dx is equal to:

Answer: (c) $$\frac { e^x }{(1+x^2)^2}$$ + c

Question 29.
$$\int_{a+c}^{b+c}$$ f(x)dx is equal to :
(a) $$\int_{c}^{b}$$ f(x – c)dx
(b) $$\int_{c}^{b}$$ f(x + c)dx
(c) $$\int_{c}^{b}$$ f(x)dx
(d) $$\int_{a-c}^{b-c}$$

Answer: (b) $$\int_{c}^{b}$$ f(x + c)dx

Question 30.
$$\int_{-1}^{1}$$ $$\frac { x^3+|x|+1 }{x^2+2|x|+1}$$ is equal to
(a) log 2
(b) 2 log 2
(c) $$\frac { 1 }{2}$$ log 2
(d) 4 log 2

Question 31.
$$\int_{c}^{b}$$ |x cos πx|dx is equal to
(a) $$\frac { 8 }{π}$$
(b) $$\frac { 4 }{π}$$
(c) $$\frac { 2 }{π}$$
(d) $$\frac { 1 }{π}$$

Answer: (a) $$\frac { 8 }{π}$$

Question 32.
If $$\int_{0}^{1}$$ $$\frac { e^t }{1+t}$$ dt = a, then $$\int_{0}^{1}$$ $$\frac { e^t }{(1+t)^2}$$
(a) a – 1 + $$\frac { e }{2}$$
(b) a + 1 – $$\frac { e }{2}$$
(c) a – 1 – $$\frac { e }{2}$$
(d) a + 1 + $$\frac { e }{2}$$

Answer: (b) a + 1 – $$\frac { e }{2}$$

Question 33.
If x = $$\int_{0}^{y}$$ $$\frac { dt }{\sqrt{1+9t^2}}$$ and $$\frac { d^y }{dx^2}$$ = ay, then a is equal to
(a) 3
(b) 6
(c) 9
(d) 1.

Question 34.
Let I = $$\int_{0}^{1}$$ $$\frac { sin x }{√x}$$ dx and J = $$\int_{0}^{1}$$ $$\frac { cos x }{√x}$$ dx. Then which of the following is true?
(a) I > $$\frac { 2 }{3}$$ and J < 2
(b) I > $$\frac { 2 }{3}$$ and J > 2
(c) I < $$\frac { 2 }{3}$$ and J < 2
(d) I < $$\frac { 2 }{3}$$ and J > 2.

Answer: (c) I < $$\frac { 2 }{3}$$ and J < 2
Hint:

Question 35.
$$\int_{0}^{π}$$ [cot x]dx, where [ . ] denotes the greatest integer function, is equal to
(a) $$\frac { π }{2}$$
(b) 2
(c) -1
(d) –$$\frac { π }{2}$$

Answer: (d) –$$\frac { π }{2}$$
Hint:

Question 36.
Let p (x) be a function defined on R such that p'(x) = p'(1 – x), for all x ∈ [0, 1], p(0) = 1 and p (1) = 41.
Then $$\int_{0}^{1}$$ p(x) dx equals
(a) $$\sqrt { 41}$$
(b) 21
(c) 41
(d) 42

Hint:
Here p'(x) – p'(1 – x).
Integrating, p (x) = -p (1 – x) + c ………… (1)
At x = 0, p(0) = -p (1) + c
⇒ 1 = -41 + c ⇒ c = 42.
Putting in (1),
p (x) = -p(1 – x) + 42
∴ $$\int_{0}^{1}$$ p(x) dx = –$$\int_{0}^{1}$$ p(1 – x)dx + $$\int_{0}^{1}$$ 42 dx
⇒ 21 = 42[x]$$_{ 0 }^{1}$$
⇒ 21 = 42
⇒ I = 21.

Question 37.
Let In = ∫tan” x dx, (n > 1).
If I4 + I6 = a tan5 x + bx5 + c, where c is a constant of integration, then the ordered pair (a, b) is equal to
(a) ($$\frac { 1}{5}$$, -1)
(b) (-$$\frac { 1}{5}$$, 0)
(c) (-$$\frac { 1}{5}$$, 1)
(d) ($$\frac { 1}{5}$$, 0)

Answer: (d) ($$\frac { 1}{5}$$, 0)
Hint:
Here I4 + I6 = a tan5 x + bx5 + c
⇒ ∫tan4x dx + ∫tan6 x dx = a tan5 x + bx5 + c.
Diff. both sides,
tan4 x + tan6 x = 5a tan4 x sec² x + 5bx4
= 5 a tan4 x(1 + tan2 x) + 5 bx4
= 5a tan4 x + 5a tan6x + 5bx4.
Comparing, 1 = 5a and 5b = 0
⇒ a = $$\frac { 1 }{5}$$ and b = 0.
Hence, (a, b) = ($$\frac { 1}{5}$$, 0)

Question 38.
The integral

Answer: (b) $$\frac {-1}{3(1+tan^3 x)}$$ + c
Hint:

Question 39.
The value of $$\int_{-π/2}^{π/2}$$ $$\frac { sin^2 x }{1 + 2^x}$$ dx is
(a) $$\frac { π}{8}$$
(b) $$\frac {π}{2}$$
(c) 4π
(d) $$\frac {π}{4}$$

Answer: (d) $$\frac {π}{4}$$
Hint:

Fill in the blanks

Question 1.
∫(√x + $$\frac {1}{√x}$$) dx (x ≠ 0) = ………………

Answer: $$\frac { 2 }{3}$$ x√x + 2√x

Question 2.
∫cot x dx = ………………….

Answer: log |sin x| + c

Question 3.
∫sec x dx = ………………

Answer: log |sec x + tan x| + c

Question 4.
∫ $$\frac { sin^2 x – cos^2 x }{sin x cos x}$$ dx = ………………..

Answer: log |sec x| – log |sin x| + c

Question 5.
∫ $$\frac { x^3+5x^2+4 }{x^2}$$ dx = ………………

Answer: $$\frac { x^2 }{2}$$ + 5x – $$\frac { 4 }{x}$$ + c

Question 6.
∫tan² x dx ………………..

Answer: tan x – x+ c

Question 7.
∫ $$\sqrt { a^2+x^2}$$ dx = ………………..

Answer: $$\frac{x \sqrt{a^{2}+x^{2}}}{2}$$ + $$\frac { a^2 }{2}$$ log|x + $$\sqrt { a^2+x^2}$$| + c

Question 8.
∫ (2 – x) sin x dx = ……………..

Answer: -2 cos x + x cos x – sin x + c

Question 9.
If ∫ ex(tan x + 1) sec x dx = ex f(x) + c, then f(x) = ………………

Question 10.
∫ $$\frac { 1 }{9x^2-1}$$ dx = …………….

Answer: $$\frac { 1 }{6}$$ log |$$\frac { 3x-1 }{3x+1}$$| + c

Question 11.
$$\int_{2}^{3}$$ 3x dx = …………………….

Answer: $$\frac { 18 }{log 3}$$

Question 12.
$$\int_{0}^{1}$$ $$\frac { dx }{\sqrt{1+x^2}}$$ = ……………..

Question 13.
If $$\int_{0}^{1}$$ (3x² + 2x + k) dx = 0, then the value of ‘k’ ………………..

Question 14.
If f(x) = $$\int_{0}^{x}$$ t sin t dt, then the value of f'(x) = ………………….

$$\int_{0}^{1.5}$$ [x] dx = ………………. where [x] is greatest integer function.