HSC Arts 12th Board ExamMaharashtra State Board
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Solution for Evaluate : ∫π0 x/(a^2cos^2 x+b^2 sin^2 x)dx - HSC Arts 12th Board Exam - Mathematics and Statistics

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Question

Evaluate : `int_0^pi(x)/(a^2cos^2x+b^2sin^2x)dx`

Solution

`I=int_0^pix/(a^2cos^2x+b^2sin^2x)dx.............(i)`

`I=int_0^pi(pi-x)/(a^2cos^2(pi-x)+b^2sin^2(pi-x))dx`

`I=int_0^pi(pi-x)/(a^2cos^2x+b^2sin^2x)dx...........(ii)`

Adding (i) and (ii), we get

`2I=int_0^pi(x)/(a^2cos^2x+b^2sin^2x)dx+int_0^pi(pi-x)/(a^2cos^2x+b^2sin^2x)dx`

`2I=int_0^pi(pi)/(a^2cos^2x+b^2sin^2x)dx`

`I=pi/2int_0^pi1/(a^2cos^2x+b^2sin^2x)dx`

`I=pi/2int_0^pisec^2x/(a^2+b^2tan^2x) dx` (dividing numerator and denominator by cos2 x)

Substitute tanx=t ⇒ sec2xdx=dt

`t=tanx=0 " at " x=0 ,t=tanx =0 `

`I=pi/2int_0^0dt/(a^2+b^2t^2)=0`

 

 

 

 

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APPEARS IN

 2015-2016 (March) (with solutions)
Question 5.1.3 | 3.00 marks
Solution Evaluate : ∫π0 x/(a^2cos^2 x+b^2 sin^2 x)dx Concept: Methods of Integration - Integration by Substitution.
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