Integrals Test

Integrals

This is Integrals Test-02 for CBSE class 12 Maths.. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

1
\(\int {{x^{ - 1}}} \) dx is equal to
  • A
    \({\log _e}\left| x \right|\)
    Correct
  • B
    \({\log _e}x\)
  • C
    \(\frac{{{x^0}}}{0}\)
  • D
    \({\log _{10}}x\)
2
If \(\int {f(x)} \) dx = f (x), then
  • A
    f (x) = \({e^x}\)
    Correct
  • B
    f (x) = 0
  • C
    f (x) = x
  • D
    b (x) constant
3
\(\int {{{\tan }^2}} \)x dx is equal to
  • A
    2 tan x \({\sec ^2}x\)
  • B
    tan x + x
  • C
    tan x – x
    Correct
  • D
    none of these.
4
\(\int {\frac{1}{{\sqrt {1 - x} }}dx} \) is equal to
  • A
    none of these
  • B
    \( - 2\sqrt {1 - x} \)
    Correct
  • C
    \(2\sqrt {1 - x} \)
  • D
    \(\sqrt {1 - x} \)
5
An antiderivative of \(\frac{x}{{{{\cos }^2}x}}\)is equal to
  • A
    None of these
  • B
    \(x\tan x + \log \left| {\cos x} \right|\)
    Correct
  • C
    Cotx
  • D
    x tanx
6
\(\int {\log } \) x dx is equal to
  • A
    x log x – x
    Correct
  • B
    none of these
  • C
    \(\frac{1}{2}{(\log x)^2}\)
  • D
    \(\frac{1}{x}\)
7
If f (x) be a function such that\(\frac{d}{{dx}}(f(x)) = \log x,\)then f (x) is equal to
  • A
    \(x\log \left( {\frac{x}{e}} \right)\)
    Correct
  • B
    \(x\log \left( {\frac{e}{x}} \right)\)
  • C
    x log (x e)
  • D
    \(\frac{{\log x}}{x}\)
8
\(\int {\frac{{f'(x)}}{{\sqrt {f(x)} }}} dx\)is equal to
  • A
    \(\frac{1}{2}\log \left| {f(x)} \right|\)
  • B
    \(2\sqrt {f(x)} \)
    Correct
  • C
    Log ( f(x) )
  • D
    f(x)
9
\(\int {\log \left( {\frac{1}{x} - x} \right)} \)dx is equal to
  • A
    log 2
  • B
    0
    Correct
  • C
    \(\frac{1}{2}\log 2\)
  • D
    none of these
10
\(\int {(\log (\log x) + {{(\log x)}^{ - 1}})} \)dx is equal to
  • A
    x log (log x) + \(\frac{x}{{\log x}}\)
  • B
    none of these.
  • C
    x log (log x) \( - \frac{x}{{\log x}}\)
  • D
    x log (log x)
    Correct
11
Which of the following is not equal to\(\int {\frac{{ - 1}}{{\sqrt {1 - {x^2}} }}} dx\)
  • A
    \( - {\sin ^{ - 1}}x\)
  • B
    \(\frac{\pi }{2} - {\cos ^{ - 1}}x\)
    Correct
  • C
    \({\cos ^{ - 1}}x\)
  • D
    \(\frac{\pi }{2} - {\sin ^{ - 1}}x\)
12
\(\int {(1 - \cos x)} \)\(\cos e{c^2}\;x\) dx is equal to
  • A
    \(2\tan \frac{x}{2} + C\)
  • B
    \(\tan \frac{x}{2} + C\)
    Correct
  • C
    \(\frac{1}{2}\tan \frac{x}{2} + C\)
  • D
    \(\cot \frac{x}{2} + C\)
13
\(\int {\sqrt {{x^2} + {a^2}} dx} \) is equal to
  • A
    Option A
  • B
    none of these
  • C
    Option C
  • D
    Option D
    Correct
14
\(\int {\frac{{\cos x}}{{(1 + \sin x)(2 + \sin x)}}dx = } \)
  • A
    \(\log \left( {\frac{{1 + \sin x}}{{2 + \sin x}}} \right) + C\)
    Correct
  • B
    \(\log \left( {\frac{{2 + \sin x}}{{1 - \sin x}}} \right) + C\)
  • C
    \(\log \left( {\frac{{2 + \sin x}}{{1 + \sin x}}} \right) + C\)
  • D
    log (1+ sin x) (2 + sin x) + C
15
\(\int {{e^x}(f(x) + f'(x))} \) is equal to
  • A
    \({e^x}(f(x) = f'(x))\)
  • B
    \({e^x}f(x)\)
    Correct
  • C
    \({e^x}f(x)\)
  • D
    none of these