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  • Can someone help me with this, - Integral Calculus - BITSAT

\int 3\sqrt{\frac{\sin^{n}x}{\cos ^{n+6}x}}.dx \,\, n\epsilon N=

  • Option 1)

    \frac{3}{n}(\tan x)^{\frac{n}{3}+1}+c

  • Option 2)

    \frac{3}{3+n}(\tan x)^{\frac{n}{3}+1}+c

  • Option 3)

    \frac{3}{n}(\cos x)^{\frac{n}{3}+1}+c

  • Option 4)

    None of these

 

Answers (1)

best_answer

As learnt

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

 

 

\int \sqrt{\frac{\sin ^{n}x}{\cos ^{n+6}x}} \: dx \: \: \: = \: \: \int \left [ \frac{\sin ^{n}n}{\cos ^{n}x\cdot \cos ^{6}x} \right ]^{\frac{1}{3}}dx

                               =\int \left ( tan^{n}x \right )^{\frac{1}{3}}\cdot \sec ^{2}xdx

Let

 \\ tanx=t \\ sec^{2}xdx=dt

\int t^{\frac{n}{3}}dt=\frac{3}{n+3}t^{\frac{n}{3}+1}

              =\frac{3}{n+3}(tanx)^{\frac{n}{3}+1}

 


Option 1)

\frac{3}{n}(\tan x)^{\frac{n}{3}+1}+c

This option is incorrect

Option 2)

\frac{3}{3+n}(\tan x)^{\frac{n}{3}+1}+c

This option is correct

Option 3)

\frac{3}{n}(\cos x)^{\frac{n}{3}+1}+c

This option is incorrect

Option 4)

None of these

This option is incorrect

Posted by

Aadil

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