∫0π2tan7xcot7x + tan7xdx is equal to from

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Integrals

Multiple Choice Questions

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821.
$\int_{0}^{\frac{π}{2}} \frac{\tan^{7} (x)}{\cot^{7} (x) + \tan^{7} (x)} dx$ is equal to

$\frac{π}{4}$

$\frac{π}{2}$

$\frac{π}{6}$

$\frac{π}{3}$

A.

$\frac{π}{4}$

$Let I = \int_{0}^{\frac{π}{2}} \frac{\tan^{7} (x)}{\cot^{7} (x) + \tan^{7} (x)} dx . . . (i) \Rightarrow I = \int_{0}^{\frac{π}{2}} \frac{\tan^{7} (\frac{π}{2} - x)}{\cot^{7} (\frac{π}{2} - x) + \tan^{7} (\frac{π}{2} - x)} dx \Rightarrow I = \int_{0}^{\frac{π}{2}} \frac{\cot^{7} (x)}{\cot^{7} (x) + \tan^{7} (x)} d x . . . (ii) On adding Eqs . (i) and (ii), we get 2 I = \int_{0}^{\frac{π}{2}} \frac{\tan^{7} (x) + \cot^{7} (x)}{\cot^{7} (x) + \tan^{7} (x)} dx = \int_{0}^{\frac{π}{2}} 1 dx = {[x]}_{0}^{\frac{π}{2}} = \frac{π}{2} ∴ I = \frac{π}{4}$

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822.

$\int {xsec}^{2} (x) dx$ is equal to

$xtan (x) + \log (\sec (x)) + c$
$\frac{x^{2}}{2} \sec (x) + \log (\cos (x)) + c$
$xtan (x) + \log (\cos (x)) + c$
$\tan (x) + \log (\cos (x)) + c$

823.

$\int \frac{t}{e^{3 t^{2}}} dt$ is equal to

$\frac{1}{6} e^{3 t^{2}} + c$
$- \frac{1}{6} e^{3 t^{2}} + c$
$\frac{1}{6} e^{- 3 t^{2}} + c$
$- \frac{1}{6} e^{- 3 t^{2}} + c$

824.

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

$2 {πlog}_{e} (\frac{1}{2})$
${πlog}_{e} (2)$
$\frac{π}{2} \log_{e} (\frac{1}{2})$
None of these

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825.

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

$\frac{1}{n} \log (\frac{x^{n}}{x^{n} + 1}) + c$
$\frac{1}{n} \log (\frac{x^{n} + 1}{x^{n}}) + c$
$\log (\frac{x^{n}}{x^{n} + 1}) + c$
None of these

826.

$\int_{0}^{2} [x^{2}] dx$ is equal to

$2 - \sqrt{2}$
$2 + \sqrt{2}$
$\sqrt{2} - 1$
$- \sqrt{2} - \sqrt{3} + 5$

827.

In = $\int_{0}^{\frac{π}{4}} \tan^{n} (x) dx$ , then $\lim_{n \to \infty} n [I_{n} + I_{n + 2}]$ equals

1/ 2
2 sq units
3 sq units
4 sq uits

828.

$\int_{0}^{10 π} |\sin (x)| dx$ is equal to

20
8
10
18

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829.

If I = $\int_{x_{0}}^{x_{0} + nh} ydx$ , then by Trapezoidal rule I is equal to

$h [(y_{0} + y_{n}) + 2 (y_{1} + y_{2} + . . . + y_{n - 1})]$
$h [\frac{1}{2} (y_{0} + y_{n}) + 2 (y_{1} + y_{2} + . . . + y_{n - 1})]$
$\frac{h}{2} [(y_{0} + y_{n}) + 2 (y_{1} + y_{2} + . . . + y_{n - 1})]$
$h [(y_{0} + y_{n}) + 2 (y_{1} + y_{2} + . . . + y_{n - 1})]$

830.

$\int \frac{dx}{1 - x^{2}}$ is equal to

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

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