Integral Calculus

$\int \frac{x}{4 + x^{4}} d x$ is equal to:
1. $\frac{1}{4} \tan^{- 1} x^{2} + C$
2. $\frac{1}{4} \tan^{- 1} (\frac{x^{2}}{2}) + C$
3. $\frac{1}{2} \tan^{- 1} (\frac{x^{2}}{2}) + C$
4. None of these
$\int \frac{1}{\cos x + \sqrt{3} \sin x} d x$ is equal to:
1. $\log \tan (\frac{x}{2} + \frac{π}{3}) + C$
2. $\log \tan (\frac{x}{2} - \frac{π}{3}) + C$
3. $\frac{1}{2} \log \tan (\frac{x}{2} + \frac{π}{3}) + C$
4. None of these
$\int x \sec x^{2} d x$ is equal to:
1. $\frac{1}{2} \log (\sec x^{2} + \tan x^{2}) + C$
2. $\frac{x^{2}}{2} \log (\sec x^{2} + \tan x^{2}) + C$
3. $2 \log (\sec x^{2} + \tan x^{2}) + C$
4. None of these
If $\int \frac{1}{5 + 4 \sin x} d x = A \tan^{- 1} (B \tan \frac{x}{2} + \frac{4}{3}) + C$ , then:
1. $A = \frac{2}{3}, B = \frac{5}{3}$
2. $A = \frac{1}{3}, B = \frac{2}{3}$
3. $A = - \frac{2}{3}, B = \frac{5}{3}$
4. $A = \frac{1}{3}, B = - \frac{5}{3}$
$\int x^{\sin x} (\frac{\sin x}{x} + \cos x . \log x) d x$ is equal to:
1. $x^{\sin x} + C$
2. $x^{\sin x} \cos x + C$
3. $\frac{{(x^{\sin x})}^{2}}{2} + C$
4. None of these
Integration of $\frac{1}{1 + {(\log_{e} x)}^{2}}$ with respect to $\log_{e} x$ is:
1. $\frac{\tan^{- 1} (\log_{e} x)}{x} + C$
2. $\tan^{- 1} (\log_{e} x) + C$
3. $\frac{\tan^{- 1} x}{x} + C$
4. None of these
If $\int \frac{\cos 8 x + 1}{\tan 2 x - \cot 2 x} d x = a \cos 8 x + C$ , then a = ?
1. $- \frac{1}{16}$
2. $\frac{1}{8}$
3. $\frac{1}{16}$
4. $- \frac{1}{8}$
If $\int \frac{\sin^{8} x - \cos^{8} x}{1 - 2 \sin^{2} x \cos^{2} x} d x = a \sin 2 x + C$ , then a = ?
1. $- \frac{1}{2}$
2. $\frac{1}{2}$
3. - 1
4. 1
$\int (x - 1) e^{- x} d x$ is equal to:
1. $- x e^{x} + C$
2. $x e^{x} + C$
3. $- x e^{- x} + C$
4. $x e^{- x} + C$
If $\int \frac{2^{\frac{1}{x}}}{x^{2}} d x = k 2^{\frac{1}{x}} + C$ , then k is equal to:
1. $- \frac{1}{\log_{e} 2}$
2. $- \log_{e} 2$
3. - 1
4. $\frac{1}{2}$
$\int \frac{1}{1 + \tan x} d x$ = ?
1. $\log_{e} (x + \sin x) + C$
2. $\log_{e} (\sin x + \cos x) + C$
3. $2 \sec^{2} \frac{x}{2} + C$
4. $\frac{1}{2} \{x + \log (\sin x + \cos x)\} + C$
$\int {|x|}^{3} d x$ is equal to:
1. $\frac{- x^{4}}{4} + C$
2. $\frac{{|x|}^{4}}{4} + C$
3. $\frac{x^{4}}{4} + C$
4. None of these
The value of $\int \frac{\cos \sqrt{x}}{\sqrt{x}} d x$ is:
1. $2 \cos \sqrt{x} + C$
2. $\sqrt{\frac{\cos x}{x}} + C$
3. $\sin \sqrt{x} + C$
4. $2 \sin \sqrt{x} + C$
$\int e^{x} (1 - \cot x + \cot^{2} x) d x$ = ?
1. $e^{x} \cot x + C$
2. $- e^{x} \cot x + C$
3. $e^{x} c o s e c x + C$
4. $- e^{x} c o s e c x + C$
$\int \frac{\sin^{6} x}{\cos^{8} x} d x$ is equal to:
1. $\tan 7 x + C$
2. $\frac{\tan^{7} x}{7} + C$
3. $\frac{\tan 7 x}{7} + C$
4. $\sec^{7} x + C$
$\int \frac{1}{7 + 5 \cos x} d x$ = ?
1. $\frac{1}{\sqrt{6}} \tan^{- 1} (\frac{1}{\sqrt{6}} \tan \frac{x}{2}) + C$
2. $\frac{1}{\sqrt{3}} \tan^{- 1} (\frac{1}{\sqrt{3}} \tan \frac{x}{2}) + C$
3. $\frac{1}{4} \tan^{- 1} (\tan \frac{x}{2}) + C$
4. $\frac{1}{7} \tan^{- 1} (\tan \frac{x}{2}) + C$
$\int \frac{1}{1 - \cos x - \sin x} d x$ = ?
1. $\log |1 + \cot \frac{x}{2}| + C$
2. $\log |1 - \tan \frac{x}{2}| + C$
3. $\log |1 - \cot \frac{x}{2}| + C$
4. $\log |1 + \tan \frac{x}{2}| + C$
$\int \frac{x - 3}{{(x + 4)}^{2}} e^{x} d x$ = ?
1. $\frac{e^{x}}{x + 4} + C$
2. $\frac{e^{x}}{x + 3} + C$
3. $\frac{1}{{(x + 4)}^{2}} + C$
4. $\frac{e^{x}}{{(x + 4)}^{2}} + C$
$\int \frac{\sin x}{3 + 4 \cos^{2} x} d x$ = ?
1. $\log (3 + 4 \cos^{2} x) + C$
2. $\frac{1}{2 \sqrt{3}} t {an}^{- 1} (\frac{\cos x}{\sqrt{3}}) + C$
3. $- \frac{1}{2 \sqrt{3}} t {an}^{- 1} (\frac{2 \cos x}{\sqrt{3}}) + C$
4. $\frac{1}{2 \sqrt{3}} t {an}^{- 1} (\frac{2 \cos x}{\sqrt{3}}) + C$
$\int e^{x} (\frac{1 - \sin x}{1 - \cos x}) d x$ = ?
1. $- e^{x} \tan \frac{x}{2} + C$
2. $- e^{x} \cot \frac{x}{2} + C$
3. $- \frac{1}{2} e^{x} \tan \frac{x}{2} + C$
4. $- \frac{1}{2} e^{x} \cot \frac{x}{2} + C$
$\int \frac{2}{{(e^{x} + e^{- x})}^{2}} d x$ is equal to:
1. $\frac{- e^{- x}}{e^{x} + e^{- x}} + C$
2. $- \frac{1}{e^{x} + e^{- x}} + C$
3. $\frac{- 1}{{(e^{x} + 1)}^{2}} + C$
4. $\frac{1}{e^{x} + e^{- x}} + C$
$\int \frac{e^{x} (1 + x)}{\cos^{2} (x e^{x})} d x$ = ?
1. $2 \log_{e} \cos (x e^{x}) + C$
2. $\sec (x e^{x}) + C$
3. None of these
4. $\tan (x e^{x}) + C$
$\int \frac{\sin^{2} x}{\cos^{4} x} d x$ is equal to:
1. $\frac{1}{3} \tan^{2} x + C$
2. $\frac{1}{2} \tan^{2} x + C$
3. $\frac{1}{3} \tan^{3} x + C$
4. None of these
The primitive of the function $f (x) = (1 - \frac{1}{x^{2}}) a^{x + \frac{1}{x}}, a > 0$ is:
1. $\frac{a^{x + \frac{1}{x}}}{\log_{e} a}$
2. $a^{x + \frac{1}{x}} . \log_{e} a$
3. $\frac{a^{x + \frac{1}{x}}}{x} \log_{e} a$
4. $x \frac{a^{x + \frac{1}{x}}}{\log_{e} a}$
The value of $\int \frac{1}{x + x \log x} d x$ is:
1. $(1 + \log x) + C$
2. $(x + \log x) + C$
3. $x \log (1 + \log x) + C$
4. $\log (1 + \log x) + C$
$\int \sqrt{\frac{x}{1 - x}} d x$ is equal to:
1. $\sin^{- 1} \sqrt{x} + C$
2. $\sin^{- 1} \{\sqrt{x} - \sqrt{x (1 - x)}\} + C$
3. $\sin^{- 1} \sqrt{x (1 - x)} + C$
4. $\sin^{- 1} \sqrt{x} - \sqrt{x (1 - x)} + C$
$\int e^{x} \{f (x) + f' (x)\} d x$ is equal to:
1. $e^{x} f (x) + C$
2. $e^{x} + f (x) + C$
3. $2 e^{x} f (x) + C$
4. $e^{x} - f (x) + C$
The value of $\int \frac{\sin x + \cos x}{\sqrt{1 - \sin 2 x}} d x$ is equal to:
1. $\sqrt{\sin 2 x} + C$
2. $\sqrt{\cos 2 x} + C$
3. $\pm (\sin x - \cos x) + C$
4. $\pm \log (\sin x - \cos x) + C$
If $\int x \sin x d x = - x \cos x + α$ , then $α$ is equal to:
1. $\sin x + C$
2. $\cos x + C$
3. C
4. None of these
$\int \frac{\cos 2 x - 1}{\cos 2 x + 1} d x$ = ?
1. $\tan x - x + C$
2. $x + \tan x + C$
3. $x - \tan x + C$
4. $- x - \cot x + C$
$\int \frac{\cos 2 x - \cos 2 θ}{\cos x - \cos θ} d x$ is equal to:
1. $2 (\sin x + x \cos θ) + C$
2. $2 (\sin x - x \cos θ) + C$
3. $2 (\sin x + 2 x \cos θ) + C$
4. $2 (\sin x - 2 x \cos θ) + C$
$\int \frac{x^{9}}{{(4 x^{2} + 1)}^{6}} d x$ is equal to:
1. $\frac{1}{5 x} {(\frac{1}{x^{2}} + 4)}^{- 5} + C$
2. $\frac{1}{5} {(\frac{1}{x^{2}} + 4)}^{- 5} + C$
3. $\frac{1}{10 x} {(\frac{1}{x^{2}} + 4)}^{- 5} + C$
4. $\frac{1}{10} {(\frac{1}{x^{2}} + 4)}^{- 5} + C$
$\int \frac{x^{3}}{\sqrt{1 + x^{2}}} d x = a {(1 + x^{2})}^{\frac{3}{2}} + b \sqrt{1 + x^{2}} + C$ , then:
1. $a = \frac{1}{3}, b = 1$
2. $a = - \frac{1}{3}, b = 1$
3. $a = - \frac{1}{3}, b = - 1$
4. $a = \frac{1}{3}, b = - 1$
$\int \frac{x^{3}}{x + 1} d x$ is equal to:
1. $x + \frac{x^{2}}{2} + \frac{x^{3}}{3} - \log |1 + x| + C$
2. $x + \frac{x^{2}}{2} - \frac{x^{3}}{3} - \log |1 + x| + C$
3. $x - \frac{x^{2}}{2} - \frac{x^{3}}{3} - \log |1 + x| + C$
4. $x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \log |1 + x| + C$
If $\int \frac{1}{(x + 2) (x^{2} + 1)} d x = a \log |1 + x^{2}| + b \tan^{- 1} x + \frac{1}{5} \log |x + 2| + C$ , then:
1. $a = - \frac{1}{10}, b = - \frac{2}{5}$
2. $a = \frac{1}{10}, b = - \frac{2}{5}$
3. $a = - \frac{1}{10}, b = \frac{2}{5}$
4. $a = \frac{1}{10}, b = \frac{2}{5}$

Integral Calculus

Quiz