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Tell me? - Integral Calculus - JEE Main

 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)
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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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