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Integrals

Multiple Choice Questions

271.

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

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

272.

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

$\frac{1}{\log (a)} \sin^{- 1} (a^{x}) + c$
$\frac{1}{\log (a)} \tan^{- 1} (a^{x}) + c$
$2 \sqrt{a^{- x} - a^{x}} + c$
$\log (a^{x} - 1) + c$

273.

If g(x) = $\frac{f (x) - f (- x)}{2}$ defined over [- 3, 3], and f(x) = 2x² - 4x + 1, then $\int_{- 3}^{3} g (x) dx$ is equal to :

274.

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

$xcos (a) - \sin (a) \log (\sin (x - a)) + c$
$xsin (a) + c$
$xsin (a) + \sin (a) \log (\sin (x - a)) + c$
$xcos (a) + \sin (a) \log (\sin (x - a)) + c$

275.

$\int \frac{f' (x)}{f (x) \log [f (x)]} dx$ is equal to :

$\frac{f (x)}{\log (f (x))}$
f(x) . log(f(x)) + c
$\log [\log (f (x))] + c$
$\frac{1}{\log [\log (f (x))]}$ + c

276.

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

$\log (\frac{1 + e^{x}}{2 + e^{x}}) + c$
$\log (\frac{2 + e^{x}}{1 + e^{x}}) + c$
$(\frac{1 + e^{x}}{2 + e^{x}}) + c$
$(\frac{2 + e^{x}}{1 + e^{x}}) + c$

277.
If I_n = $\int_{0}^{\frac{π}{4}} \tan^{n} (θ) dθ$ , then $I_{θ} + I_{6}$ is equal to :

$\frac{1}{7}$

$\frac{1}{4}$

$\frac{1}{5}$

$\frac{1}{6}$

$\frac{1}{7}$

$∵ I_{n} = \int_{0}^{\frac{π}{4}} \tan^{n} (θ) dθ ∴ I_{8} + I_{6} = \int_{0}^{\frac{π}{4}} \tan^{8} (θ) dθ + \int_{0}^{\frac{π}{4}} \tan^{6} (θ) dθ = \int_{0}^{\frac{π}{4}} \sec^{2} (θ) \tan^{6} (θ) dθ = {[\frac{\tan^{7} (θ)}{7}]}_{0}^{\frac{π}{4}} = \frac{1}{7}$