Integrals Test

Integrals

This is Integrals Test-04 for CBSE class 12 Maths.. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

1
\(\int\limits_0^2 {\left[ {{x^2}} \right]} dx\) equals
  • A
    none of these
  • B
    3
  • C
    \(\frac{8}{2}\)
  • D
    \(5 - \sqrt 3 - \sqrt 2 \)
    Correct
2
\(\int\limits_1^e {\frac{{1 + \log x}}{{2x}}} dx\) is equal to
  • A
    \(\frac{3}{4}\)
    Correct
  • B
    \(\frac{1}{4}\)
  • C
    \(\frac{1}{e}\)
  • D
    e
3
\(\int\limits_0^{\pi /6} {\frac{{\cos 2x}}{{{{(\cos x - \sin x)}^2}}}dx} \)
  • A
    none of these
  • B
    \( - \log \left( {\frac{{\sqrt 3 + 1}}{2}} \right)\)
  • C
    \( - \log \left( {\frac{{\sqrt 3 - 1}}{2}} \right)\)
    Correct
  • D
    \(\log \left( {\frac{{\sqrt 3 + 1}}{2}} \right)\)
4
\(\int {{{\tan }^{ - 1}}} \left( {\frac{{3x - {x^3}}}{{1 - 3{x^2}}}} \right)\) dx can be evaluated by the substitution
  • A
    x = sin \(\theta \)
  • B
    x = tan \(\theta \)
    Correct
  • C
    x = cos\(\theta \)
  • D
    x = sec \(\theta \)
5
\(\int {\frac{{\cos (\log x)}}{x}dx} \) is equal to
  • A
    sin(log x)
    Correct
  • B
    none of these
  • C
    – sin (log x)
  • D
    \(\frac{{\sin (\log x)}}{x}\)
6
\(\int {\frac{1}{{{e^x} + 1}}dx} \) is equal to
  • A
    none of these
  • B
    – log \((1 + {e^{ - x}})\)
    Correct
  • C
    \(\log ({e^{ - x}} + 1)\)
  • D
    log \((1 + {e^{ - x}})\)
7
\(\int {\frac{1}{{1 + 3{{\sin }^2}x}}dx} \) is equal to
  • A
    \(\frac{1}{2}{\tan ^{ - 1}}(2\tan x)\)
    Correct
  • B
    \({\tan ^{ - 1}}(2\tan x)\)
  • C
    \(\frac{1}{2}{\tan ^{ - 1}}\left( {\frac{{\tan x}}{2}} \right)\)
  • D
    none of these
8
\(\int {\frac{{\sec x\cos ecx}}{{\log (\tan x)}}dx} \;is\;equal\)
  • A
    log | log (tan x) |
    Correct
  • B
    tan {log (log x) }
  • C
    log (tan x)
  • D
    tan (log x)
9
\(\int {\frac{1}{{{x^2}}}\sin \left( {\frac{1}{x}} \right)} dx\)is equal to
  • A
    \(\cos \left( {\frac{1}{x}} \right) + C\)
    Correct
  • B
    \( - \cos \left( {\frac{1}{{{x^2}}}} \right) + C\)
  • C
    \( - \cos \left( {\frac{1}{x}} \right) + C\)
  • D
    \(\cos \left( {\frac{1}{{{x^2}}}} \right) + C\)
10
\(\int {\frac{{{x^2} - 1}}{{{x^4} + 3{x^2} + 1}}} dx\) is equal to
  • A
    \({\tan ^{ - 1}}\left( {x + \frac{1}{x}} \right) + C\)
    Correct
  • B
    \(\tan \left( {x + \frac{1}{x}} \right) + C\)
  • C
    \({\tan ^{ - 1}}\left( {{x^2} + 1} \right) + C\)
  • D
    none of these.
11
.\(\int\limits_{ - 1/2}^{1/2} {\cos x\log \left( {\frac{{1 + x}}{{1 - x}}} \right)} dx\). is equal to
  • A
    0
    Correct
  • B
    1
  • C
    – 1
  • D
    none of these
12
\(\int\limits_0^2 {\left[ {2x} \right]} \) is equal to
  • A
    4
  • B
    2
  • C
    none of these
  • D
    3
    Correct
13
\(\int\limits_{ - 8}^8 {({{\sin }^{93}}x + {x^{295}})} \) dx is equal to
  • A
    \(2 + {8^{295}}\)
  • B
    0
    Correct
  • C
    a number different from 0
  • D
    \(2({8^{295}} + 1)\)
14
\(\int\limits_0^{\pi /2} {\frac{1}{{1 + \sqrt {\cot x} }}dx} \) is equal to
  • A
    none of these
  • B
    \(\frac{\pi }{4}\)
    Correct
  • C
    \(\pi \)
  • D
    \(\frac{\pi }{4}\)
15
\(\int\limits_{ - \pi /12}^{\pi /12} {\frac{1}{{\cos 2x}}dx = } \)
  • A
    none of these.
  • B
    log 3
  • C
    \(\frac{1}{3}\log 2\)
  • D
    \(\frac{1}{2}\log 3\)
    Correct