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int(0)^(pi//2)(sinxcosx)/((1+sin^(4)x))d...

`int_(0)^(pi//2)(sinxcosx)/((1+sin^(4)x))dx`

Text Solution

Verified by Experts

Put `sin^(2)x=t`.
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int_(0)^(pi//2)(sin x.cos x)/(1+sin^(4)x)dx=

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Evaluate the following integral: int_0^(pi//2)(sinxcosx)/(1+sin^4x)dx

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int_(0)^((pi)/(2))(sin x cos x)/(1+sin^(4)x)dx

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int_(0)^(pi//2) (cosx )/ (1+sin^(2)x)dx=

I_(1)=int_(0)^((pi)/2)(sinx-cosx)/(1+sinxcosx)dx, I_(2)=int_(0)^(2pi)cos^(6)dx , I_(3)=int_(-(pi)/2)^((pi)/2)sin^(3)xdx, I_(4)=int_(0)^(1) In (1/x-1)dx . Then