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Integrals

Multiple Choice Questions

561.

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

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

562.
If I_n = $\int_{0}^{\frac{π}{4}} \tan^{n} (θ) dθ$ , then $I_{θ} + I_{6}$ is equal to :

$\frac{1}{7}$

$\frac{1}{4}$

$\frac{1}{5}$

$\frac{1}{6}$

$\frac{1}{7}$

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

563.

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

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

564.

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

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

565.

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

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

566.

If f(x) = $\cos (x) - \cos^{2} (x) + \cos^{3} (x) - . . . to \infty$ , then $\int f (x) dx$ equals to

$\tan (\frac{x}{2}) + c$
$x + \tan (\frac{x}{2}) + c$
$x - \frac{1}{2} \tan (\frac{x}{2}) + c$
$x - \tan (\frac{x}{2}) + c$

567.

$\int_{0}^{1} \frac{xdx}{[x + \sqrt{1 - x^{2}}] \sqrt{1 - x^{2}}}$ is equal to :

0
1
$\frac{π}{4}$
$\frac{π^{2}}{2}$

568.

The value of $\int_{0}^{2 a} \frac{f (x) dx}{f (x) + f (2 a - x)}$ is :

f(a)
f(2a)
f(0)
a

569.

The value of $\int_{- \frac{π}{4}}^{\frac{π}{4}} x^{3} \sin^{4} (x) dx$ is equal to :

$\frac{π}{4}$
$\frac{π}{2}$
$\frac{π}{8}$
0

570.

For any positive integer n, $\int \frac{dx}{x^{n + 1} + x}$ is equal to :

$\frac{1}{n} \log_{e} (x^{n} + 1) + c$
$\frac{1}{n} \log_{e} (\frac{1}{x^{n} + 1}) + c$
$\frac{1}{n} \log_{e} (\frac{x}{x^{n} + 1}) + c$
$\frac{1}{n} \log_{e} (\frac{x^{n}}{x^{n} + 1}) + c$

Previous Year Papers

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Test Yourself

Multiple Choice Questions

562. If In = ∫0π4tannθdθ, then Iθ + I6 is equal to : 17 14 15 16

562.
If I_n = $\int_{0}^{\frac{π}{4}} \tan^{n} (θ) dθ$ , then $I_{θ} + I_{6}$ is equal to :

$\frac{1}{7}$

$\frac{1}{4}$

$\frac{1}{5}$

$\frac{1}{6}$