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Integrals

Multiple Choice Questions

201.

$\int \frac{e^{x}}{x} (xlog (x) + 1) dx$ is equal to

$\frac{e^{x}}{x} + C$
${xe}^{x} \log |x| + C$
$e^{x} \log |x| + C$
x(e^x + $\log |x|$ ) + C

202.

$\int \frac{1 + \log (x)}{{(1 + x \log (x))}^{2}}$ dx is equal to

$\frac{1}{1 + xlog |x|} + C$
$\frac{1}{1 + \log |x|}$ + C
$\frac{- 1}{1 + xlog |x|} + C$
$\log |\frac{1}{1 + \log |x|}|$ + C

203.

$\int (1 - \tan^{2} (x)) dx$ is equal to

$\tan (x) + C$
$\sec (x) + C$
$2 x - \sec (x) + C$
2x - $\tan (x)$ + C

204.

The value of $\int_{0}^{6} |x - 3| d x$ is equal to

205.

If f(x) = $\int_{2 x}^{\sin (x)} \cos (t^{3}) dt$ , then f'(x) is equal to

$\cos (\sin^{3} (x)) \cos (x) - 2 \cos (8 x^{3})$
$\sin (\sin^{3} (x)) \sin (x) - 2 \sin (8 x^{3})$
$\cos (\cos^{3} (x)) \cos (x) - 2 \cos (x^{3})$
$\cos (\sin^{3} (x)) - \cos (8 x^{3})$

206.

$\int_{0}^{10} \frac{x^{10}}{{(10 - x)}^{10} + x^{10}} dx$ is equal to

10
5
2
$\frac{1}{2}$

207.

The value of $\int_{0}^{1} \sqrt{x} e^{\sqrt{x}} dx$ is equal to

$\frac{(e - 2)}{2}$
2(e - 2)
2e - 1
2(e - 1)

208.

$\int \frac{1}{x (\log (x)) \log (\log (x))} dx$ is equal to

$\log (\log (x)) + C$
$\log |\log (xlog (x))| + C$
$\log |\log |\log (\log (x))|| + C$
$\log |\log (\log (x))| + C$

209.
$\int \frac{3^{x}}{\sqrt{1 - 9^{x}}}$ dx is equal to

$\log (3) \sin^{- 1} (3^{x}) + C$

$\frac{1}{3} \sin^{- 1} (3^{x}) + C$

$(\frac{1}{3}) \sin^{- 1} (3^{x}) + C$

$\frac{1}{9} \sin^{- 1} (3^{x}) + C$

$(\frac{1}{3}) \sin^{- 1} (3^{x}) + C$

$Let I = \int \frac{3^{x}}{\sqrt{1 - 9^{x}}} dx \Rightarrow I = \int \frac{3^{x}}{\sqrt{1 - {(3^{x})}^{2}}} dx Put t = 3^{x}, dt = 3^{x} \log (3) dx Then, I = \frac{1}{\log (3)} \int \frac{dt}{\sqrt{1 - t^{2}}} \Rightarrow I = \frac{1}{\log (3)} . \sin^{- 1} (t) + C \Rightarrow I = (\frac{1}{\log (3)}) . \sin^{- 1} (3^{x}) + C$