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Let $In =\int \tan ^{n}\: x\: dx,\left ( n>1 \right )$

If $I_{4}+I_{6}= a \tan ^{5}\: x+bx^{5} +C$ where C is a constant of integration, then the ordered pair (a, b) is equal to :

Option 1)
$\left ( \frac{1}{5} ,0\right )$

Option 2)
$\left ( \frac{1}{5} ,-1\right )$

Option 3)
$\left ( -\frac{1}{5} ,0\right )$

Option 4)
$\left ( -\frac{1}{5},1 \right )$

Answers (1)

best_answer

As learnt in concept

Integration of trigonometric function of power m -

$\int tan^{m}xdx , \int cot^{m}xdx$

- wherein

for $m=2$

use $tan^{2}x=sec^{2}x-1$ , $cot^{2}x=cosec^{2}x-1$

$I_{n} =\int \left ( \tan ^{n} \: x\right )dx$

$I_{4}+I_{6}=\int \left ( \tan ^{4}x+\tan ^{6}x \right )dx$

$=\int \tan ^{4}x\left ( 1+\tan ^{2}x \right )dx$

$=\int \tan ^{4}x.\sec ^{2}x\: dx$

$=\frac{\tan ^{5}x}{5}+C$

Here $a=\frac{1}{5}$

b = 0

Option 1)

$\left ( \frac{1}{5} ,0\right )$

Correct option

Option 2)

$\left ( \frac{1}{5} ,-1\right )$

Incorrect option

Option 3)

$\left ( -\frac{1}{5} ,0\right )$

Incorrect option

Option 4)

$\left ( -\frac{1}{5},1 \right )$

Incorrect option

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prateek

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