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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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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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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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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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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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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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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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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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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Question 10.
\(\int_{-2}^{2}\) |x|dx =
(a) 0
(b) 2
(c) 1
(d) 4

Answer

Answer: (d) 4

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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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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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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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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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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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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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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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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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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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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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