Quiz Tweek

Integrals MCQ Set-1

Test your knowledge on Integrals from Mathematics, Class 12.

100

Minutes

60

Questions

1 / -0

Marking Scheme

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Questions in this Quiz

Q1: $\int \frac{1}{2x-1} dx$
  • $\frac{1}{2} \log|2x-1| + C$
  • $\log|2x-1| + C$
  • $-\frac{1}{2} \log|2x-1| + C$
  • $2 \log|2x-1| + C$
Q2: $\int (3\sin^2 x + \cos^2 x) dx$
  • $2x – \frac{1}{2}\sin 2x + C$
  • $x + \sin 2x + C$
  • $2x + \frac{1}{2}\cos 2x + C$
  • $x – \cos 2x + C$
Q3: $\int e^{3-2x} dx$
  • $e^{3-2x} + C$
  • $\frac{1}{2} e^{3-2x} + C$
  • $-\frac{1}{2} e^{3-2x} + C$
  • $-\frac{1}{3} e^{3-2x} + C$
Q4: $\int \sin(e^x) e^x dx$
  • $\cos(e^x) + C$
  • $-\cos(e^x) + C$
  • $\sin(e^x) + C$
  • $e^x \cos(e^x) + C$
Q5: $\int e^{\log x^{-2}} dx$
  • $\frac{1}{x} + C$
  • $-\frac{1}{x} + C$
  • $\frac{x^{-3}}{-3} + C$
  • $\frac{x^2}{2} + C$
Q6: $\int (\sec^2 x - \csc^2 x) dx$
  • $\tan x - \cot x + C$
  • $\tan x + \cot x + C$
  • $\tan x + 2\cot x + C$
  • $2\tan x + \cot x + C$
Q7: $\int \frac{\sin 2x}{1 + \cos 2x} dx$
  • $-\log|\cos x| + C$
  • $\log|\sec x| + C$
  • $\frac{1}{2} \log|\sec x| + C$
  • $-\frac{1}{2} \log|\cos x| + C$
Q8: $\int \frac{\cos 2x - 1}{\cos 2x + 1} dx$
  • $\tan x - x + C$
  • $x - \tan x + C$
  • $\cot x - x + C$
  • $x - \cot x + C$
Q9: $\int \frac{\cos 2x}{\cos x} dx$
  • $2\sin x - \log|\sec x + \tan x| + C$
  • $\sin x - \log|\sec x + \tan x| + C$
  • $2\sin x + \log|\sec x + \tan x| + C$
  • $2\cos x - \log|\sec x + \tan x| + C$
Q10: $\int \frac{1}{(3-x)^2} dx$
  • $-\frac{1}{3-x} + C$
  • $\frac{1}{3-x} + C$
  • $-\log|3-x| + C$
  • $\frac{-1}{(3-x)^3} + C$
Q11: $\int \sin^{-1} (\cos x) dx$
  • $\frac{\pi x}{2} + \frac{x^2}{2} + C$
  • $\frac{\pi}{2} - x + C$
  • $\frac{\pi x}{2} - \frac{x^2}{2} + C$
  • $\frac{\pi x}{2} - x^2 + C$
Q12: $\int e^x \left(\frac{x-1}{x^2}\right) dx$
  • $\frac{e^x}{x^2} + C$
  • $\frac{e^x}{x} + C$
  • $e^x \log|x| + C$
  • $-\frac{e^x}{x} + C$
Q13: $\int \frac{dx}{\sqrt{1-4^x}}$
  • $\frac{1}{\log 2} \sin^{-1}(2^x) + C$
  • $\sin^{-1}(2^x) + C$
  • $\frac{1}{\log 4} \sin^{-1}(2^x) + C$
  • $\frac{2}{\log 2} \sin^{-1}(2^x) + C$
Q14: $\int e^{\tan^{-1} \sqrt{x}} \frac{1}{\sqrt{x} + x \sqrt{x}} dx$
  • $e^{\tan^{-1} \sqrt{x}} + C$
  • $2 e^{\tan^{-1} \sqrt{x}} + C$
  • $\frac{1}{2} e^{\tan^{-1} \sqrt{x}} + C$
  • $e^{\tan^{-1} x} + C$
Q15: $\int \frac{e^{x}+e x^{e-1}}{e^{x}+x^{e}} d x$
  • $\frac{1}{e} \log|e^x + x^e| + C$
  • $\log|e^x + x^e| + C$
  • $e \log|e^x + x^e| + C$
  • $\log|e^x + x^{e-1}| + C$
Q16: $\int \frac{1}{1 + e^x} dx$
  • $\log|1+e^x| + C$
  • $\log|e^x + 1| - x + C$
  • $x - \log|e^x + 1| + C$
  • $\frac{1}{2} \log|1+e^x| + C$
Q17: If $\int \frac{1+x^2}{1+x^3} dx = \log|x| + f(x) + C$, then $f'(x)$ is:
  • $\frac{1}{x+x^5}$
  • $x^5$
  • $\frac{x^3}{1+x^5}$
  • $\frac{x^4}{x+x^5}$
Q18: $\int \tan 3x dx$
  • $\frac{1}{3} \log|\sec 3x| + C$
  • $3 \log|\sec 3x| + C$
  • $\log|\sec 3x| + C$
  • $-\frac{1}{3} \log|\cos 3x| + C$
Q19: $\int a^{\sqrt{x}} \frac{1}{\sqrt{x}} dx$
  • $\frac{2 a^{\sqrt{x}}}{\log a} + C$
  • $\frac{a^{\sqrt{x}}}{\log a} + C$
  • $2 a^{\sqrt{x}} \log a + C$
  • $a^{\sqrt{x}} \frac{\sqrt{x}}{2} + C$
Q20: $\int \cot^2 x \csc^2 x dx$
  • $\frac{1}{3} \cot^3 x + C$
  • $-\frac{1}{3} \cot^3 x + C$
  • $\cot^3 x + C$
  • $\frac{1}{2} \cot^2 x + C$
Q21: $\int \tan^6 x \sec^2 x dx$
  • $\frac{1}{7} \tan^7 x + C$
  • $\frac{1}{6} \tan^6 x + C$
  • $\tan^7 x + C$
  • $\frac{1}{7} \sec^7 x + C$
Q22: $\int \frac{x}{1+x^4} dx$
  • $\frac{1}{2} \tan^{-1}(x^2) + C$
  • $\tan^{-1}(x^2) + C$
  • $\log|1+x^4| + C$
  • $\frac{x^2}{1+x^4} + C$
Q23: $\int \frac{\cot x}{\sqrt{\sin x \cos x}} dx$
  • $-2 \sqrt{\sin x} + C$
  • $-2 \sqrt{\cos x} + C$
  • $2 \sqrt{\sin x} + C$
  • $-2 \sqrt{\sin x} + C$
Q24: $\int \frac{\cos x}{a+b \sin x} dx$
  • $\frac{1}{b} \log|a+b \sin x| + C$
  • $b \log|a+b \sin x| + C$
  • $\frac{1}{a} \log|a+b \sin x| + C$
  • $\log|a+b \sin x| + C$
Q25: $\int \frac{1}{e^{-x} + 1} dx$
  • $x + \log|1+e^x| + C$
  • $\log|1+e^x| + C$
  • $x - \log|1+e^x| + C$
  • $\frac{1}{2} \log|1+e^x| + C$
Q26: $\int \frac{\cos^3 x}{\sin^2 x + \sin x} dx$
  • $\log|\sin x| - \sin x + C$
  • $\log|\sin x| - \cos x + C$
  • $\log|\sin x| - \sin x + \cos x + C$
  • $\log|\sin x| + \sin x + C$
Q27: $\int \frac{\cos x - \sin x}{\sin x + \cos x} dx$
  • $\log|\sin x + \cos x| + C$
  • $-\log|\sin x + \cos x| + C$
  • $\frac{1}{2} \log|\sin x + \cos x| + C$
  • $\sin x + \cos x + C$
Q28: $\int e^x (\cot x - \csc^2 x + 1) dx$
  • $e^x (\cot x) + C$
  • $e^x (1 + \cot x) + C$
  • $e^x (\cot x - 1) + C$
  • $e^x (\csc^2 x) + C$
Q29: $\int \frac{1-x}{\sqrt{x}} dx$
  • $2\sqrt{x} - \frac{2}{3} x^{3/2} + C$
  • $\sqrt{x} - x^{3/2} + C$
  • $\frac{1}{2\sqrt{x}} - \frac{1}{2\sqrt{x}} + C$
  • $x^{1/2} - x^{3/2} + C$
Q30: $\int x \sin x dx$
  • $-x \cos x - \sin x + C$
  • $x \cos x - \sin x + C$
  • $-x \cos x + \sin x + C$
  • $x \sin x + \cos x + C$
Q31: $\int \frac{\cos 2x}{(\sin x + \cos x)^2} dx$
  • $\frac{-1}{\sin x + \cos x} + C$
  • $\log_e|\sin x + \cos x| + C$
  • $\log_e|\sin x - \cos x| + C$
  • $\frac{1}{(\sin x + \cos \theta)^2} + C$
Q32: $\int \frac{dx}{\sin (x-a) \sin (x-b)}$
  • $\sin (b-a) \log |\frac{\sin (x-b)}{\sin (x-a)}| + C$
  • $\csc (b-a) \log |\frac{\sin (x-a)}{\sin (x-b)}| + C$
  • $\csc (b-a) \log |\frac{\sin (x-b)}{\sin (x-a)}| + C$
  • $\sin (b-a) \log |\frac{\sin (x-a)}{\sin (x-b)}| + C$
Q33: $\int \tan^{-1} \sqrt{x} dx$
  • $(x + 1)\tan^{-1}\sqrt{x} - \sqrt{x} + C$
  • $x \tan^{-1}\sqrt{x} - \sqrt{x} + C$
  • $\sqrt{x} - x \tan^{-1}\sqrt{x} + C$
  • $\sqrt{x} - (x + 1)\tan^{-1}\sqrt{x} + C$
Q34: $\int \frac{\sin^6 x}{\cos^8 x} dx$
  • $\frac{1}{5} \tan^6 x + C$
  • $\frac{1}{5} \tan^7 x + C$
  • $\frac{1}{7} \tan^7 x + C$
  • None of these
Q35: $\int \frac{x}{(x-1)(x-2)} dx$
  • $\log_e|\frac{(x-1)^2}{x-2}| + C$
  • $\log_e|\frac{(x-2)^2}{x-1}| + C$
  • $\log_e|(\frac{x-1}{x-2})^2| + C$
  • $\log_e|\frac{x-1}{x-2}| + C$
Q36: $\int (x^2 + 2x + 2)^{-1} dx$
  • $x \tan^{-1} (x + 1) + C$
  • $\tan^{-1} (x + 1) + C$
  • $(x + 1) \tan^{-1}x + C$
  • $x \tan^{-1}x + C$
Q37: $\int_{0}^{\pi/2} \sqrt{1 - \sin 2x} dx$
  • $2\sqrt{2}$
  • $2(\sqrt{2} + 1)$
  • 2
  • $2(\sqrt{2} - 1)$
Q38: The anti-derivative of $(\sqrt{x} + 1/\sqrt{x})$ is
  • $\frac{1}{3} x^{1/3} + 2x^{1/2} + C$
  • $\frac{2}{3} x^{2/3} + \frac{1}{2} x^2 + C$
  • $\frac{2}{3} x^{3/2} + 2x^{1/2} + C$
  • $\frac{3}{2} x^{3/2} + \frac{1}{2} x^{1/2} + C$
Q39: If $f’(x) = 4x^3 - 3/x^4$ such that $f(2) = 0$. Then $f(x)$ is
  • $x^4 + x^{-3} - \frac{129}{8}$
  • $x^3 + x^{-4} + \frac{129}{8}$
  • $x^4 + x^{-3} + \frac{129}{8}$
  • $x^3 + x^{-4} - \frac{129}{8}$
Q40: $\int_{1}^{2} x^{2} d x$
  • 1
  • $\frac{7}{3}$
  • $\frac{1}{3}$
  • 0
Q41: $\int_{0}^{2}\left(x^{2}+3\right) d x$
  • $\frac{25}{3}$
  • $\frac{26}{3}$
  • $\frac{24}{3}$
  • None of these
Q42: Evaluate: $\int_{0}^{\pi / 4} \sqrt{1-\sin 2 x} d x$
  • $\sqrt{2} – 1$
  • $\sqrt{2} + 1$
  • $\sqrt{2}$
  • None of these
Q43: Evaluate: $\int_{0}^{2 \pi} \sin \left(\frac{\pi}{4}+\frac{x}{2}\right) d x$
  • $-2\sqrt{2}$
  • $-2$
  • $\sqrt{2}$
  • $2\sqrt{2}$
Q44: Evaluate: $\int_{1}^{2} \frac{d x}{x^{2}}$
  • $\frac{1}{2}$
  • 1
  • 2
  • -1
Q45: $\int (2\tan x – 3\cot x)^2 dx$
  • $-4\tan x – 9\cot x – 25x + C$
  • $4\tan x – 9\cot x – 25x + C$
  • $-4\tan x + 9 \cot x + 25x + C$
  • $4\tan x + 9\cot x + 25x + C$
Q46: $\int e^x (\cos x - \sin x) dx$
  • $e^x \cos x + c$
  • $e^x \cos x - c$
  • $-e^x \cos x - c$
  • $-e^x \cos x + c$
Q47: $\int_{-\pi/2}^{\pi/2} (x^3 + x \cos x + \tan^5 x + 1) dx$
  • 0
  • 1
  • $\pi$
  • $2\pi$
Q48: If $f(a+b-x) = f(x)$, then $\int_{a}^{b} x f(x) d x$ is equal to:
  • $\frac{a-b}{2} \int_{a}^{b} f(x) d x$
  • $(a+b) \int_{a}^{b} f(x) d x$
  • $\frac{a+b}{2} \int_{a}^{b} f(x) d x$
  • $\frac{a}{2} \int_{a}^{b} f(x) d x$
Q49: If $f(x) = \int_{0}^{x} t \sin t dt$, then $f'(x)$ is:
  • $\sin x$
  • $t \sin t$
  • $x \sin x$
  • $x \cos x$
Q50: $\int_{-\pi/4}^{\pi/4} \frac{1}{1 + \cos 2x} dx$
  • 0
  • $\frac{1}{2}$
  • 1
  • 2
Q51: $\int_{0}^{2/3} \frac{1}{4 + 9x^2} dx$
  • $\frac{\pi}{12}$
  • $\frac{\pi}{24}$
  • $\frac{\pi}{6}$
  • $\frac{\pi}{36}$
Q52: $\int \frac{10x^9 + 10^x \log_e 10}{x^{10} + 10^x} dx$
  • $\log |10^x| + C$
  • $\log |x^{10} + 10^x| + C$
  • $\frac{1}{2} \log |x^{10} + 10^x| + C$
  • $e^x (x^{10} + 10^x) + C$
Q53: $\int_{0}^{\pi/2} \log \left( \frac{4+3\sin x}{4+3\cos x} \right) dx$
  • 1
  • $\frac{\pi}{2}$
  • $\log 3$
  • 0
Q54: $\int_{0}^{1} \tan^{-1} \left( \frac{2x-1}{1+x-x^2} \right) dx$
  • $\pi/4$
  • 1
  • 0
  • $\pi/2$
Q55: $\int \tan^{-1} \sqrt{x} dx$
  • $x \tan^{-1} \sqrt{x} + \sqrt{x} + \tan^{-1} \sqrt{x} + C$
  • $x \tan^{-1} \sqrt{x} - \sqrt{x} + \tan^{-1} \sqrt{x} + C$
  • $\sqrt{x} \tan^{-1} \sqrt{x} + x + C$
  • $\frac{1}{\sqrt{x}} \tan^{-1} \sqrt{x} + C$
Q56: $\int e^x \left(\frac{x+3}{(x+4)^2}\right) dx$
  • $e^x \frac{x+3}{x+4} + C$
  • $e^x \frac{1}{x+4} + C$
  • $e^x \log|x+4| + C$
  • $e^x \frac{x+4}{x+3} + C$
Q57: $\int \frac{x^3}{x+1} dx$
  • $\frac{x^4}{4} - \frac{x^3}{3} + x^2 - x + \log|1+x| + C$
  • $x^2 - x + \log|1+x| + C$
  • $x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\log |1+x|+C$
  • $\frac{x^4}{4} + \log|1+x| + C$
Q58: The value of $\int e^{x}\left(\frac{x^{2}+2 x+1}{\left(1+x^{2}\right)^{2}}\right) d x$ is:
  • $\frac{e^{x}}{1+x^{2}}+C$
  • $\frac{e^{x}}{x^{2}-1}+C$
  • $\frac{e^{x}}{x+1}+C$
  • $\frac{e^{x}}{x-1}+C$
Q59: $\int e^x \left(\frac{x+2}{x+4}\right) dx$
  • $e^{x}\left(\frac{x+1}{x+4}\right)+C$
  • $e^{x}\left(\frac{x+2}{x+4}\right)+C$
  • $e^{x}\left(\frac{x+3}{x+4}\right)+C$
  • $e^{x}\left(\frac{x+4}{x+2}\right)+C$
Q60: $\int \frac{\cos 2x - \cos 2\theta}{\cos x - \cos \theta} dx$
  • $2(\sin x + 2x \cos \theta) + C$
  • $2(\sin x - x \cos \theta) + C$
  • $2(\sin x + x \cos \theta) + C$
  • $\sin x - x \cos \theta + C$