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Integrals

Multiple Choice Questions

541.

$\int_{0}^{x} f (t) dt = x^{2} + e^{x}$ (x > 0), then f(1) is equal to

1 + e
2 + e
3 + e
e

542.

$\int \frac{x + 1}{x^{\frac{1}{2}}} dx =$

- x^3/2 + x^1/2 + C
x^1/2
x^3/2 + 2x^1/2 + C
$\frac{2 x^{\frac{3}{2}}}{3} + 2 x^{\frac{1}{2}} + C$

543.

$\int \frac{dx}{e^{x} + e^{- x} + 2}$ is equal to

$\frac{1}{e^{x} + 1} + C$
$\frac{- 1}{e^{x} + 1} + C$
$\frac{1}{1 + e^{- x}} + C$
$\frac{1}{e^{- x} - 1} + C$

544.

$\int_{- 1}^{0} \frac{dx}{x^{2} + x + 2}$ is equal to

$\frac{π}{4}$
$\frac{π}{2}$
$π$
$\frac{4}{\sqrt{7}} \tan^{- 1} (\frac{1}{\sqrt{7}})$

545.

$\int_{- 2}^{2} |1 - x^{2}| dx$ is equal to :

546.

$\int \frac{\sqrt{\tan (x)}}{\sin (x) \cos (x)} dx$ is equal to :

$2 \tan (x) + c$
$\sqrt{\cot (x)} + c$
$2 \sqrt{\tan (x)} + c$
$\tan^{2} (x) + c$

547.

$\int \frac{dx}{x^{2} + 4 x + 13}$ is equal to :

$\log (x^{2} + 4 x + 130) + c$
$\frac{1}{3} \tan^{- 1} (\frac{x + 2}{3}) + c$
$\log (2 x + 4)$ + c
$\frac{1}{x^{2} + 4 x + 13} + c$

548.

$\int_{0}^{1} \frac{d}{dx} [\sin^{- 1} (\frac{2 x}{1 + x^{2}})] dx$ is equal to :

0
$π$
$\frac{π}{2}$
$\frac{π}{4}$

549.
$\int_{0}^{\frac{π}{2}} \frac{\sin (x)}{\sin (x) + \cos (x)} dx$ is equal to

$π$

$\frac{π}{2}$

$\frac{π}{3}$

$\frac{π}{4}$

$\frac{π}{4}$

$Let I = \int_{0}^{\frac{π}{2}} \frac{\sin (x)}{\sin (x) + \cos (x)} dx . . . (i) I = \int_{0}^{\frac{π}{2}} \frac{\sin (\frac{π}{2} - x)}{\sin (\frac{π}{2} - x) + \cos (\frac{π}{2} - x)} dx I = \int_{0}^{\frac{π}{2}} \frac{\cos (x)}{\cos (x) + \sin (x)} dx . . . (ii) On adding Eqs . (i) and (ii), we get 2 I = \int_{0}^{\frac{π}{2}} 1 dx = \frac{π}{2} \Rightarrow I = \frac{π}{4}$