Integrals Test

Integrals

This is Integrals Test-01 for CBSE class 12 Maths.. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

1
If \(\int {f(x)dx = g(x),} \) then
  • A
    g (x) + f (x) =constant.
  • B
    \(\frac{d}{{dx}}(f(x)) = g(x)\)
  • C
    g (x) = f (x)
  • D
    \(\frac{d}{{dx}}(g(x)) = f(x)\)
    Correct
2
\(\frac{d}{{dx}}(\int {f(x)dx)} \) is equal to
  • A
    none of these.
  • B
    \(\frac{{(f{{(x)}^2}}}{2}\)
  • C
    f (x)
    Correct
  • D
    f (x)
3
\(\int {{e^{ - \log x}}dx} \) is equal to
  • A
    none of these
  • B
    \(\log \left| x \right|\)
    Correct
  • C
    \( - {e^{ - \log x}}\)
  • D
    \( - x{e^{ - \log x}}\)
4
\(\int {\tan x} {\sec ^2}\) x dx is not equal to
  • A
    none of these
    Correct
  • B
    \(\frac{1}{2}{(\cos x)^{ - 2}}\)
  • C
    \(\frac{{{{\sec }^2}x}}{2}\)
  • D
    \(\frac{{{{\tan }^2}x}}{2}\)
5
If \(\int {f(x)} \) dx = g (x) and also \(\int {f(x)} \) dx = h (x), then
  • A
    h (x) + g (x) = constant
  • B
    g (x) = h (x).
  • C
    g (x) - h (x) = constant
    Correct
  • D
    g (x) h (s) = constant
6
One value of\(\int {f'(x)} \) dx is
  • A
    none of these
  • B
    \(\frac{{\int {f(x)} {)^2}}}{2}\)
  • C
    f (x)
    Correct
  • D
    f ‘ (x)
7
If \(\int {g(x)dx = g(x),} \) then\(\int {g(x)} \left( {\frac{{1 + \sin x}}{{1 + \cos x}}} \right)\) dx is equal to
  • A
    \(\frac{1}{2}g(x){\sec ^2}\frac{x}{2}\)
  • B
    \(\frac{{g(x)}}{{1 + \cos x}}\)
  • C
    g (x) tan \(\frac{x}{2}\)
    Correct
  • D
    none of these
8
\(\int {{{\sec }^2}x} \)\(\cos e{c^2}x\)dx is equal to
  • A
    none of these.
  • B
    tan x . cot x
  • C
    tan x + cotx
  • D
    tan x – cot x
    Correct
9
\(\int {\frac{{\log {x^n}}}{x}} dx,n \in N,\) is equal to
  • A
    \(\frac{{{{(\log x)}^2}}}{{2n}} + C\)
  • B
    \(n{(\log x)^2} + C\)
  • C
    \(\frac{{n{{(\log x)}^2}}}{2} + C\)
    Correct
  • D
    none of these
10
\(\int {(\tan x + cot\;x)} \) dx is equal to
  • A
    \(log{\text{ }}\left| {{\text{ }}sin{\text{ }}x{\text{ }} + {\text{ }}cos{\text{ }}x} \right|\)
  • B
    \(\begin{array}{*{20}{l}} {log{\text{ }}\left| {{\text{ cot }}x} \right|} \end{array}\)
  • C
    \(\begin{array}{*{20}{l}} {log{\text{ }}\left| {{\text{ }}tan{\text{ }}x} \right|} \end{array}\)
    Correct
  • D
    none of these
11
\(\int {\frac{{\sin 2x}}{{3 + 4{{\sin }^2}x}}dx} \)is equal to
  • A
    \(\frac{1}{4}\log (3 + 4{\sin ^2}x)\)
    Correct
  • B
    \(\frac{2}{3}\log \left( {\tan \frac{x}{2}} \right)\)
  • C
    none of these
  • D
    \(\frac{2}{9}{\tan ^{ - 1}}\left( {3\tan \frac{x}{2}} \right)\)
12
\(\int {\frac{{\cos x}}{{\cos (x - a)}}} \) dx is equal to
  • A
    sin a – sin a log | cos ( x – a) |
  • B
    none of these.
  • C
    x cos a – sin a log | cos ( x – a) |
  • D
    x cos a + sin a log | cos (x – a) |
    Correct
13
Which of the following is different from \(\int {\frac{1}{{1 + {x^2}}}dx?} \)
  • A
    \(\frac{\pi }{2} - {\cot ^{ - 1}}x\)
  • B
    \( - {\cot ^{ - 1}}x\)
  • C
    \({\tan ^{ - 1}}x\)
  • D
    none of these
    Correct
14
\(\int {\frac{{x\cos x}}{{\sqrt {x\sin x + \cos x} }}dx\;is\;equal\;to} \)
  • A
    \(2{(x\;\sin x + \cos \;x)^{1/2}} + C\)
    Correct
  • B
    \(\frac{2}{3}{(x\sin x + \cos x)^{3/2}} + C\)
  • C
    \(\frac{1}{2}{(x\sin x + \cos x)^{1/2}} + C\)
  • D
    none of these
15
\(\int {\sqrt {{a^2} - {x^2}} } \) dx is equal to
  • A
    none of these
  • B
    \(\frac{x}{2}\sqrt {{a^2} - {x^2}} - \frac{{{a^2}}}{2}{\sin ^{ - 1}}\left( {\frac{x}{a}} \right)\)
  • C
    \(\frac{x}{2}\sqrt {{a^2} - {x^2}} + \frac{{{a^2}}}{2}\log \left| {x + \sqrt {{a^2} - {x^2}} } \right|\)
  • D
    \(\frac{x}{2}\sqrt {{a^2} - {x^2}} + \frac{{{a^2}}}{2}{\sin ^{ - 1}}\left( {\frac{x}{a}} \right)\)
    Correct