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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 :

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

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

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

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

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

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

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 :

$\frac{1}{7}$
$\frac{1}{4}$
$\frac{1}{5}$
$\frac{1}{6}$

278.

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

$\log (\tanh (x))$
$2 \log (e^{x} + e^{- x}) + c$
$2 \log (e^{x} - e^{- x}) + c$
$loglog (\cosh (x)) + c$

279.

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \cos (x) \log (\frac{1 + x}{1 - x}) dx = k . \log (2)$ , then k equals to

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

280.

$\int_{0}^{\frac{π}{2}} \frac{\cos (θ)}{\sqrt{4 - \sin^{2} (θ)}} dθ$ is equal to :

$\frac{π}{2}$
$\frac{π}{6}$
$\frac{π}{3}$
$\frac{π}{5}$

Previous Year Papers

Test Series

Test Yourself

Multiple Choice Questions

272. ∫ax2a- x - axdx is equal to : 1logasin-1ax + c 1logatan-1ax + c 2a- x - ax + c logax - 1 + c

272.
$\int \frac{a^{\frac{x}{2}}}{\sqrt{a^{- x} - a^{x}}} dx$ is equal to :

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

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

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

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