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Integrals

Multiple Choice Questions

531.

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

$\log |\tan (x)| + C$
$\log |\sin (2 x)| + C$
$\log |\sec (x)| + C$
$\log |\cos (x)| + C$

532.

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

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

533.

$\int \frac{1}{8 \sin^{2} (x) + 1} dx$ is equal to

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

534.

$\int_{0}^{\frac{π}{2}} \log (\frac{\cos (x)}{\sin (x)}) dx$ is equal to

$\frac{π}{2}$
$\frac{π}{4}$
$π$
0

535.

The value of $\int_{- 1}^{2} 4 x^{2} |x| d x$ is equal to

536.

The value of $\int_{2}^{4} (x - 2) (x - 3) (x - 4) dx$ is equal to

$\frac{1}{2}$
2
3
0

537.

$\int_{0}^{\frac{π}{2}} \frac{\sqrt{\sin (x)}}{\sqrt{\sin (x)} + \sqrt{\cos (x)}} dx$ is equal to

0
$- π$
$\frac{3 π}{2}$
$\frac{π}{4}$

538.
$\int_{2016}^{2017} \frac{\sqrt{x}}{\sqrt{x} + \sqrt{4033} - x} dx$ is equal to

1/4

3/2

2017/2

1/2

1/2

$Let I = \int_{2016}^{2017} \frac{\sqrt{x}}{\sqrt{x} + \sqrt{4033} - x} dx . . . (i) ∴ I = \int_{2016}^{2017} \frac{\sqrt{4033 - x}}{\sqrt{4033 - x} + \sqrt{4033 - (4033 - x)}} dx [∵ \int_{a}^{b} f (x) dx = \int_{a}^{b} f (a + b - x) dx] \Rightarrow I = \int_{2016}^{2017} \frac{\sqrt{4033 - x}}{\sqrt{4033 - x} + \sqrt{x}} dx . . . (ii) On adding Eqs . (i) and (ii), we get 2 I = \int_{2016}^{2017} dx \Rightarrow 2 I = {[x]}_{2016}^{2017} \Rightarrow 2 I = 1 \Rightarrow I = \frac{1}{2}$