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Integrals

Multiple Choice Questions

261.

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

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

262.

$\int \frac{dx}{x^{2} + 4 x + 13}$ 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
$\frac{1}{x^{2} + 4 x + 13} + c$

263.

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

0
$π$
$\frac{π}{2}$
$\frac{π}{4}$

264.

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

$π$
$\frac{π}{2}$
$\frac{π}{3}$
$\frac{π}{4}$

265.

$\int \frac{\sin (x)}{\sin (x - α)} dx$ is equal to

$(x - α) \cos (α) + \sin (α) \log (\sin (x - α)) + c$
$\sin (x - α) + \sin (x) + c$
$(x - α) \cos (α) + \log (\sin (x - α)) + c$
$\cos (x - α) + \cos (x) + c$

266.

$\int 13^{x} dx$ is :

$\frac{13^{x}}{\log (13)} + c$
$13^{x + 1} + c$
14x + c
14^{x + 1} + c

267.

$\int_{0}^{\frac{π}{2}} \sin (2 x) \log (\tan (x)) d x$

$π$
$\frac{π}{2}$
1
0

268.

$\int_{0}^{\frac{π}{2}} xsin (x) dx$ is equal to :

$\frac{π}{4}$
$\frac{π}{2}$
$π$
1

269.
$\int x^{2} {(ax + b)}^{- 2} dx$ is equal to :

$\frac{2}{a^{2}} (x - \frac{b}{a} \log (ax + b))$ + c

$\frac{2}{a^{2}} (x - \frac{b}{a} \log (ax + b)) - \frac{x^{2}}{a (ax + b)}$

$\frac{2}{a^{2}} (x + \frac{b}{a} \log (ax + b)) - \frac{x^{2}}{a (ax + b)} + c$

$\frac{2}{a^{2}} (x + \frac{b}{a} \log (ax + b)) - \frac{x^{2}}{a (ax + b)} + c$

$\frac{2}{a^{2}} (x - \frac{b}{a} \log (ax + b)) - \frac{x^{2}}{a (ax + b)}$

$Let I = \int \frac{x^{2}}{{(ax + b)}^{2}} dx Put ax + b = t \Rightarrow dx = \frac{1}{a} dt and x = (\frac{t - b}{a}) ∴ I = \frac{1}{a^{3}} \int \frac{{(t - b)}^{2}}{t^{2}} dt = \frac{1}{a^{3}} \int (1 + \frac{b^{2}}{t^{2}} - \frac{2 b}{t}) dt = \frac{1}{a^{3}} (t - \frac{b^{2}}{t} - 2 blog (t)) + c = \frac{1}{a^{3}} (ax + b - \frac{b^{2}}{ax + b} - 2 blog (ax + b)) + c = \frac{2}{a^{2}} (x - \frac{b}{a} \log (ax + b)) - \frac{x^{2}}{a (ax + b)} + c$