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Evaluate: int0^(pi//2)(cos^2x)/(cos^2x+4...

Evaluate: `int_0^(pi//2)(cos^2x)/(cos^2x+4sin^2x)dx`

Text Solution

Verified by Experts

Let I=`int_0^(pi//2)(cos^2x)/(cos^2x+4sin^2x)dx`
⇒I=`int_0^(pi//2)(cos^2x)/(cos^2x+4(1-cos^2x)dx`
​ ⇒`I=−1/3​int_0^(pi//2)(4−3cos^2x-4)/(4−3cos^2x)​dx`
⇒`I=−1/3​int_0^(pi//2)(4−3cos^2x)/(4−3cos^2x)​dx+1/3​int_0^(pi//2)(4)/(4−3cos^2x)​dx`
⇒`I=−1/3int ​_0^(π/2)​​1dx+1/3​int_0^(pi//2)(4sec^2x)/(4sec^2x−3)​dx`
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Knowledge Check

  • The value of int_0^(pi//4)(sinx+cosx)/(cos^2x+sin^4x)dx is

    A
    `1/2log((sqrt3-1))/sqrt2`
    B
    `1/(2sqrt3)log((sqrt3+1)/sqrt2)+pi/4`
    C
    `1/2log((sqrt3+1)/sqrt2)+pi/4`
    D
    `1/sqrt3log((sqrt3+1)/sqrt2)+pi/4`

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