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Integrals

Multiple Choice Questions

551.

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

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

552.

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

$π$
$\frac{π}{2}$
1
0

553.

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

$\frac{π}{4}$
$\frac{π}{2}$
$π$
1

554.

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

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

555.

If f(t) is an odd function, then $\int_{0}^{x} f (t) dt$ is :

an odd function
an even function
neither even nor odd
0

556.

$\int e^{- \log (x)} dx$ is equal to :

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

557.
$\int \frac{a^{\frac{x}{2}}}{\sqrt{a^{- x} - a^{x}}} dx$ is equal to :

$\frac{1}{\log (a)} \sin^{- 1} (a^{x}) + c$

$\frac{1}{\log (a)} \tan^{- 1} (a^{x}) + c$

$2 \sqrt{a^{- x} - a^{x}} + c$

$\log (a^{x} - 1) + c$

$\frac{1}{\log (a)} \sin^{- 1} (a^{x}) + c$

$Let I = \int \frac{a^{\frac{x}{2}}}{\sqrt{a^{- x} - a^{x}}} dx = \int \frac{a^{x}}{\sqrt{1 - a^{2 x}}} dx Let a^{x} = t \Rightarrow a^{x} \log (a) dx = dt ∴ I = \int \frac{dt}{\sqrt{1 - t^{2}}} . \frac{1}{\log (a)} = \frac{1}{\log (a)} \sin^{- 1} (t) + c = \frac{1}{\log (a)} \sin^{- 1} (a^{x}) + c$