Integrals Class 12 MCQ Online Test With Answers Questions

Check the below NCERT MCQ Questions for Class 12 Maths Chapter 7 Integrals with Answers Pdf free download. MCQ Questions for Class 12 Maths with Answers were prepared based on the latest exam pattern. We have provided Integrals Class 12 Maths MCQs Questions with Answers to help students understand the concept very well.

Class 12 Maths Chapter 7 MCQ With Answers

Maths Class 12 Chapter 7 MCQs On Integrals

Integration MCQ Class 12 Chapter 7 Question 1.
\(\int \frac{x+\sin x}{1+\cos x}\) dx is equal to
(a) log |1 + cos x | + c
(b) log | x + sin x | + c
(c) x – tan + c
(d) x. tan \(\frac{x}{2}\) + c

Answer

Answer: (d) x. tan \(\frac{x}{2}\) + c

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MCQ On Integration Class 12 Chapter 7 Question 2.
∫1.dx =
(a) x + k
(b) 1 + k
(c) \(\frac{x^2}{2}\) + k
(d) log x + k

Answer

Answer: (a) x + k

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Integration MCQ Questions Class 12 Chapter 7 Question 3.
∫\(\frac{dx}{√x}\) =
(a) √x + k
(b) 2√x + k
(c) x + k
(d) \(\frac{2}{3}\)x^3/2 + k

Answer

Answer: (b) 2√x + k

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MCQ On Integration Class 12 Chapter 7 Question 4.
∫\(\frac{dx}{1+cos x}\) =
(a) tan \(\frac{x}{2}\) + k
(b) \(\frac{1}{2}\) tan \(\frac{x}{2}\) + k
(c) 2 tan \(\frac{x}{2}\) + k
(d) tan² \(\frac{x}{2}\) + k

Answer

Answer: (a) tan \(\frac{x}{2}\) + k

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Integration MCQs With Answers Pdf Class 12 Chapter 7 Question 5.
\(\int_{a}^{b}\) x⁵ dx =
(a) tan \(\frac{x}{2}\) + k
(b) \(\frac{1}{2}\) tan \(\frac{x}{2}\) + k
(c) 2 tan \(\frac{x}{2}\) + k
(d) tan² \(\frac{x}{2}\) + k

Answer

Answer: (a) tan \(\frac{x}{2}\) + k

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Integration MCQ With Answers Class 12 Chapter 7 Question 6.
If x > a, ∫\(\frac{dx}{x^2-a^2}\) =
(a) \(\frac{2}{2a}\) log \(\frac{x-a}{x+a}\) + k
(b) \(\frac{2}{2a}\) log \(\frac{x+a}{x-a}\) + k
(c) \(\frac{1}{a}\) log(x² – a²) + k
(d) log(x + \(\sqrt{x^2-a^2}\) + k)

Answer

Answer: (a) \(\frac{2}{2a}\) log \(\frac{x-a}{x+a}\) + k

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Integration Objective Questions Class 12 Chapter 7 Question 7.
∫\(\frac{cos 2x dx}{(sinx+cosx)^2}\) =
(a) –\(\frac{1}{sinx+cosx}\) + c
(b) log | sin x + cos x | + c
(c) log | sin x – cos x | + c
(d) \(\frac{1}{(sinx+cosx)^2}\)

Answer

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

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MCQ On Integration With Answers Class 12 Chapter 7  Question 8.
∫\(\frac{(1+logx)^2}{1+x^2}\) dx =
(a) \(\frac{1}{3}\)(1+log)³ + c
(b) \(\frac{1}{2}\)(1+log)² + c
(c) log (log 1 + x) + 2
(d) None of these

Answer

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

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Integration MCQs Class 12 Chapter 7  Question 9.
\(\int_{0}^{1}\frac{(tan^{-1}x)^2}{1+x^2}\) dx =
(a) 1
(b) \(\frac{π^2}{64}\)
(c) \(\frac{π^2}{192}\)
(d) None of these

Answer

Answer: (c) \(\frac{π^2}{192}\)

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MCQ Of Integration Class 12 Chapter 7 Question 10.
\(\int_{-2}^{2}\) |x|dx =
(a) 0
(b) 2
(c) 1
(d) 4

Answer

Answer: (d) 4

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MCQs On Integration Class 12 Chapter 7 Question 11.
∫\(\frac{x^4+1}{x^2+1}\) dx is equal to
(a) \(\frac{x^3}{3}\) + x + tan^-1 x + c
(b) \(\frac{x^3}{3}\) – x + tan x + c
(c) \(\frac{x^3}{3}\) + x + 2tan^-1 x + c
(d) \(\frac{x^3}{3}\) – x + 2tan^-1 x + c

Answer

Answer: (d) \(\frac{x^3}{3}\) – x + 2tan^-1 x + c

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Integration MCQ With Answers Pdf Class 12 Chapter 7 Question 12.
∫(√x + \(\frac{1}{√x}\)) dx =
(a) \(\frac{1}{x}\) x^{\(\frac{1}{3}\)} + 2x^{\(\frac{1}{2}\)} + c
(b) \(\frac{2}{3}\) x^{\(\frac{2}{3}\)} + \(\frac{1}{2}\)x² + c
(c) \(\frac{2}{3}\) x^{\(\frac{3}{2}\)} + 2x^{\(\frac{1}{2}\)} + c
(d) \(\frac{3}{2}\) x^{\(\frac{3}{2}\)} + \(\frac{1}{2}\)x^{\(\frac{1}{2}\)} + c

Answer

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

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Integration MCQ Questions And Answers Class 12 Chapter 7  Question 13.
∫\(\frac{sin^2x-cos^2x}{sin^2xcos^2x}\) dx is equal to
(a) tan x + cos x + c
(b) tan x + cosec x + c
(c) tan x + cot x + c
(d) tan x+ sec x + c

Answer

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

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Integral MCQ Questions Class 12 Chapter 7 Question 14.
\(\frac{d}{dx}\)∫f(x)dx is equal to
(a) f'(x)
(b) f(x)
(c) f'(x’)
(d) f(x) + c

Answer

Answer: (b) f(x)

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MCQ Integration Class 12 Chapter 7 Question 15.
∫\(\frac{xdx}{(x-1)(x-2)}\) equals

(d) log |(x – 1)(x – 2) + c

Answer

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

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Integration MCQs With Answers Chapter 7  Question 16.
What is the value of \(\int_{0}^{\pi / 2}\) \(\frac{\sqrt{tan x}}{\sqrt{tan x}+\sqrt{cot x}}\) dx?
(a) \(\frac{π}{2}\)
(b) \(\frac{π}{4}\)
(c) \(\frac{π}{8}\)
(d) None of these

Answer

Answer: (b) \(\frac{π}{4}\)

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Integrals MCQs Class 12 Chapter 7 Question 17.
What is the value of \(\int_{0}^{\pi / 2}\) \(\frac{sinx – cos x}{1+sin xcos x}\) dx?
(a) 1
(b) \(\frac{π}{2}\)
(c) 0
(d) –\(\frac{π}{2}\)

Answer

Answer: (c) 0

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Integration MCQs Class 12 Chapter 7 Question 18.
What is the value of \(\int_{\pi / 6}^{\pi / 3}\) \(\frac{dx}{sin2x}\)?
(a) \(\frac{1}{2}\) log(-l)
(b) log(- 1)
(c) log 3
(d) log √3

Answer

Answer: (c) log 3

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Integration MCQ Class 12 Question 19.
What is the value of \(\int_{-1}^{1}\) sin³ x cos² xdx?
(a) 0
(b) 1
(c) \(\frac{1}{2}\)
(d) 2

Answer

Answer: (a) 0

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MCQ Questions On Integration Question 20.
What is the value of \(\int_{1}^{e} \frac{1+\log x}{x}\) dx?
(a) \(\frac{3}{2}\)
(b) \(\frac{1}{2}\)
(c) e
(d) \(\frac{1}{e}\)

Answer

Answer: (a) \(\frac{3}{2}\)

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Question 21.
\(\int_{-\pi / 2}^{\pi / 2}\) sin⁹ xdx =
(a) -1
(b) 0
(c) 1
(d) None of these

Answer

Answer: (b) 0

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Question 22.
\(\int_{0}^{\pi^{2} / 4} \frac{\sin \sqrt{y}}{\sqrt{y}}\)
(a) 1
(b) 2
(c) \(\frac{π}{4}\)
(d) \(\frac{π^2}{8}\)

Answer

Answer: (b) 2

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Question 23.
\(\int_{0}^{\infty} \frac{1}{1+e^{x}}\) dx =
(a) log 2
(b) -log 2
(c) log 2 – 1
(d) log 4 – 1

Answer

Answer: (a) log 2

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Question 24.
\(\int_{0}^{1}\) x(1 – x)⁹⁹ is equal to
(a) \(\frac{1}{10010}\)
(b) \(\frac{1}{10100}\)
(c) \(\frac{1}{1010}\)
(d) \(\frac{11}{10100}\)

Answer

Answer: (b) \(\frac{1}{10100}\)

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Question 25.
What is the value of \(\int_{0}^{1}\) \(\frac{d}{dx}\){sin^-1(\(\frac{2x}{1+x^2}\))}dx?
(a) 0
(b) π
(c) -π
(d) \(\frac{π}{2}\)

Answer

Answer: (d) \(\frac{π}{2}\)

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Question 26.
\(\int_{0}^{1}\) \(\frac{x}{1+x}\) dx =
(a) 1 – log 2
(b) 2
(c) 1 + log 2
(d) log 2

Answer

Answer: (a) 1 – log 2

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Question 27.
∫\(\frac{sin x + cos x}{\sqrt{1+2sin x}}\) dx =
(a) log(sin x – cos x)
(b) x
(c) log x
(d) log sin (cos x)

Answer

Answer: (b) x

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Question 28.
∫log₁₀ xdx =
(a) log_e 10.x log_e (\(\frac{x}{e}\)) + c
(b) log₁₀ e.x log_e (\(\frac{x}{e}\)) + c
(c) (x – 1) log_e x + c
(d) \(\frac{1}{x}\) + c

Answer

Answer: (b) log₁₀ e.x log_e (\(\frac{x}{e}\)) + c

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Question 29.
∫(\(\frac{cos 2θ – 1}{cos 2θ + 1}\)) dθ =
(a) tan θ – θ + c
(b) θ + tan θ + c
(c) θ – tan θ + c
(d) -θ – cot θ + c

Answer

Answer: (c) θ – tan θ + c

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Question 30.
∫\(\frac{2dx}{\sqrt{1-4x^2}}\) =
(a) tan^-1 (2x) + c
(b) cot^-1 (2x) + c
(c) cos^-1 (2x) + c
(d) sin^-1 (2x) + c

Answer

Answer: (d) sin^-1 (2x) + c

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Question 31.
Value of ∫\(\frac{dx}{\sqrt{2x – x^2}}\)
(a) sin^-1 (x – 1) + c
(b) sin^-1 (1 + x) + c
(c) sin^-1 (1 + x²) + c
(d) –\(\sqrt{2x-x^2}\) + c

Answer

Answer: (a) sin^-1 (x – 1) + c

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Question 32.
∫x² sin x³ dx =
(a) \(\frac{1}{3}\) cos x³ + c
(b) –\(\frac{1}{3}\) cos x + c
(c) \(\frac{-1}{3}\) cos x³ + c
(d) \(\frac{1}{2}\) sin² x³ + c

Answer

Answer: (c) \(\frac{-1}{3}\) cos x³ + c

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Question 33.
∫\(\frac{cos 2x- cos 2θ}{cos x – cos θ}\)dx is equal to
(a) 2 (sin x + x cos θ) + C
(b) 2 (sin x – x cos θ) + C
(c) 2 (sin x + 2x cos θ) + C
(d) 2 (sin x – 2x cos θ) + C

Answer

Answer: (a) 2 (sin x + x cos θ) + C

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Question 34.
∫\(\frac{dx}{sin(x-a)sin(x-b)}\) is equal to
(a) sin(b – a) log |\(\frac{sin (x-b)}{sin(x-a)}\)| + C
(b) cosec (b – a) log |\(\frac{sin (x-b)}{sin(x-b)}\)| + C
(c) cosec (b – a) log |\(\frac{sin (x-b)}{sin(x-a)}\)| + C
(d) sin (b – a) log |\(\frac{sin (x-a)}{sin(x-b)}\)| + C

Answer

Answer: (c) cosec (b – a) log |\(\frac{sin (x-b)}{sin(x-a)}\)| + C

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Question 35.
∫tan^-1 √xdx 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) √x – (x + 1)tan^-1 √x + C

Answer

Answer: (a) (x + 1)tan^-1 √x – √x + C

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Question 36.
∫e^x(\(\frac{1-x}{1+x^2}\))² dx is equal to
(a) \(\frac{e^x}{1+x^2}\) + C
(b) –\(\frac{-e^x}{1+x^2}\) + C
(c) \(\frac{e^x}{(1+x^2)^2}\) + C
(d) \(\frac{-e^x}{(1+x^2)^2}\) + C

Answer

Answer: (a) \(\frac{e^x}{1+x^2}\) + C

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Question 37.
∫\(\frac{x^9}{(4x^2+1)^6}\) dx is equal to

Answer

Answer: (d) \(\frac{1}{10}\) (\(\frac{1}{x^2}\) + 4)^-5 + C

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Question 38.
If ∫\(\frac{dx}{(x+2)(x^2+1)}\) = a log |1 + x²| + b tan^-1 x + \(\frac{1}{5}\) log |x + 2| + C, then
(a) a = \(\frac{-1}{10}\), b = \(\frac{-2}{5}\)
(b) a = \(\frac{1}{10}\), b = \(\frac{-2}{5}\)
(c) a = \(\frac{-1}{10}\), b = \(\frac{2}{5}\)
(d) a = \(\frac{1}{10}\), b = \(\frac{2}{5}\)

Answer

Answer: (c) a = \(\frac{-1}{10}\), b = \(\frac{2}{5}\)

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Question 39.
∫ \(\frac{x^3}{x+1}\) is equal to
(a) x + \(\frac{x^2}{2}\) + \(\frac{x^3}{3}\) – log |1 – x| + C
(b) x + \(\frac{x^2}{2}\) – \(\frac{x^3}{3}\) – log |1 – x| + C
(c) x + \(\frac{x^2}{2}\) – \(\frac{x^3}{3}\) – log |1 + x| + C
(d) x + \(\frac{x^2}{2}\) + \(\frac{x^3}{3}\) – log |1 + x| + C

Answer

Answer: (d) x + \(\frac{x^2}{2}\) + \(\frac{x^3}{3}\) – log |1 + x| + C

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Question 40.
If ∫\(\frac{x^3dx}{\sqrt{1+x^2}}\) = a(1 + x²)^3/2 + b\(\sqrt{1 + x^2}\) + C, then
(a) a = \(\frac{1}{3}\), b = 1
(b) a = \(\frac{-1}{3}\), b = 1
(c) a = \(\frac{-1}{3}\), b = -1
(d) a = \(\frac{1}{3}\), b = -1

Answer

Answer: (d) a = \(\frac{1}{3}\), b = -1

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Question 41.
\(\int_{-\pi / 4}^{\pi / 4}\) \(\frac{dx}{1+cos 2x}\) dx is equal to
(a) 1
(b) 2
(c) 3
(d) 4

Answer

Answer: (a) 1

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Question 42.
\(\int_{0}^{\pi / 2}\) \(\sqrt{1+sin 2x}\) dx is equal to
(a) 2√2
(b) 2(√2 + 1)
(c) 0
(d) 2(√2 – 1)

Answer

Answer: (c) 0

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Question 43.
Evaluate: ∫(2 tan x – 3 cot x)² dx
(a) -4tan x – cot x – 25x + C
(b) 4 tan x – 9 cot x – 25x + C
(c) – 4 tan x + 9 cot x + 25x + C
(d) 4 tan x + 9 cot x + 25x + C

Answer

Answer: (b) 4 tan x – 9 cot x – 25x + C

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Question 44.
Evaluate: ∫ sec²(7 – 4x)dx
(a) –\(\frac{1}{4}\) tan(7 – 4x) + C
(b) \(\frac{1}{4}\) tan(7 – 4x) + C
(c) \(\frac{1}{4}\) tan(7 + 4x) + C
(d) –\(\frac{1}{4}\) tan(7x – 4) + C

Answer

Answer: (a) –\(\frac{1}{4}\) tan(7 – 4x) + C

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Question 45.
∫ \(\frac{10x^9+10^xlog_e 10}{10^x+x^{10}}\) dx is equal to
(a) 10^x – x¹⁰ + C
(b) 10^x + x¹⁰ + C
(c) (10^x – x¹⁰)^-1 + C
(d) log_e(10^x + x¹⁰) + C

Answer

Answer: (d) log_e(10^x + x¹⁰) + C

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Question 46.
Evaluate: ∫ sec^4/3 x cosec^8/3 xdx
(a) \(\frac{3}{5}\) tan^-5/3 x – 3 tan^1/3 x + C
(b) –\(\frac{3}{5}\) tan^-5/3 x + 3 tan^1/3 + C
(c) –\(\frac{3}{5}\) tan^-05/3 x – 3 tan^1/3 + C
(d) None of these

Answer

Answer: (b) –\(\frac{3}{5}\) tan^-5/3 x + 3 tan^1/3 + C

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Question 47.
∫ \(\frac{a}{(1+x^2)tan^{-1}x}\) dx =
(a) a log |tan^-1 x| + C
(b) \(\frac{1}{2}\)(tan^-1 x)² + C
(c) a log (1 + x²) + C
(d) None of these

Answer

Answer: (a) a log |tan^-1 x| + C

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Question 48.
∫ \(\frac{cot x}{\sqrt[3]{sin x}}\) dx =
(a) \(\frac{-3}{\sqrt[3]{sin x}}\) + C
(b) \(\frac{-2}{sin^3 x}\) + C
(c) \(\frac{3}{sin^{1/3}x}\) + C
(d) None of these

Answer

Answer: (a) \(\frac{-3}{\sqrt[3]{sin x}}\) + C

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Question 49.
Evaluate: ∫ \(\frac{1}{1+3sin^2x+8cos^2x}\) dx
(a) \(\frac{1}{6}\) tan^-1 (2 tan x) + C
(b) tan^-1 (2 tan x) + C
(c) \(\frac{1}{6}\) tan^-1(\(\frac{2 tan x}{3}\)) + C
(d) None of these

Answer

Answer: (c) \(\frac{1}{6}\) tan^-1(\(\frac{2 tan x}{3}\)) + C

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Question 50.
Evaluate: ∫ \(\frac{1}{\sqrt{9+8x-x^2}}\) dx
(a) -sin^-1 (\(\frac{x-4}{5}\)) + C
(b) sin^-1 (\(\frac{x+4}{5}\)) + C
(c) sin^-1 (\(\frac{x-4}{5}\)) + C
(d) None of these

Answer

Answer: (c) sin^-1 (\(\frac{x-4}{5}\)) + C

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Question 51.
∫ \(\frac{dx}{1-cosx-sinx}\) is equal to
(a) log |1 + cot\(\frac{x}{2}\)| + C
(b) log |1 – tan\(\frac{x}{2}\)| + C
(c) log |1 – cot\(\frac{x}{2}\)| + C
(d) log |1 + tan\(\frac{x}{2}\)| + C

Answer

Answer: (c) log |1 – cot\(\frac{x}{2}\)| + C

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Question 52.
Evaluate: ∫ \(\frac{1}{\sqrt{1-e^{2x}}}\) dx
(a) log |e^-x + \(\sqrt{e^{-2x} – 1}\)| + C
(b) -log |e^-x + \(\sqrt{e^{-2x} – 1}\)| + C
(c) -log |e^-x – \(\sqrt{e^{-2x} – 1}\)| + C
(d) None of these

Answer

Answer: (b) -log |e^-x + \(\sqrt{e^{-2x} – 1\)| + C

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Question 53.
If ∫ \(\frac{3x+4}{x^3-2x-4}\) dx = log |x – 2| + k log f(x) + c, then
(a) f(x) = |x² + 2x + 2|
(b) f(x) = x² + 2x + 2
(c) k = –\(\frac{1}{2}\)
(d) All of these

Answer

Answer: (d) All of these

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Question 54.
Evaluate: ∫ \(\frac{1-cosx}{cosx(1+cosx)}\) dx
(a) log|sec x + tan x| – 2 tan(x/2) + C
(b) log|sec x – tan x| – 2 tan(x/2) + C
(c) log|sec x + tan x| + 2 tan(x/2) + C
(d) None of these

Answer

Answer: (a) log|sec x + tan x| – 2 tan(x/2) + C

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Question 55.
∫ cos(log_e.x)dx is equal to
(a) \(\frac{1}{2}\) x[cos (log_ex) + sin(log_ex)]
(b) x[cos (log_ex) + sin(log_ex)]
(c) \(\frac{1}{2}\) x[cos (log_ex) – sin(log_ex)]
(d) x[cos (log_ex) – sin(log_ex)]

Answer

Answer: (b) –\(\frac{3}{5}\) tan^-5/3 x + 3 tan^1/3 + C

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Question 56.
∫ |x| dx is equal to
(a) \(\frac{1}{2}\) x² + C
(b) –\(\frac{x^2}{2}\) + C
(c) x|x| + C
(d) \(\frac{1}{2}\) x|x| + C

Answer

Answer: (d) \(\frac{1}{2}\) x|x| + C

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Question 57.
∫ sin^-1 xdx is equal to
(a) cos^-1 x + C
(b) x sin^-1x + \(\sqrt{1-x^2}\) + C
(c) \(\frac{1}{\sqrt{1-x^2}}\) + C
(d) x sin^-1x – \(\sqrt{1-x^2}\) + C

Answer

Answer: (b) x sin^-1x + \(\sqrt{1-x^2}\) + C

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Question 58.
∫ cos^-1(\(\frac{1}{x}\))dx equals
(a) x sec^-1 x + log |x + \(\sqrt{x^2-1}\)| + C
(b) x sec^-1 x – log |x + \(\sqrt{x^2-1}\)| + C
(c) -x sec^-1 x – log |x + \(\sqrt{x^2-1}\)| + C
(d) None of these

Answer

Answer: (b) x sec^-1 x – log |x + \(\sqrt{x^2-1}\)| + C

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