Evaluate the following integral: `int_0^pisin^3x(1+2cos)(1+cos x)^2dx`

Explanation

Given,I=`int_0^pisin^3x(1+2cos)(1+cos x)^2dx`

Substitute cosx=t, so that

I=`int_(-1)^1 (1-t^2)(1+2t)(1+t)^2dt`

=`2int_(-1)^1 (1-t^2)(1+5t^2)dt`

[∵ remaining part of integrand is an odd function]


=`int_0^1 ​(1+4t^2−5t^4)dt`

=`2(t+4/3​t^3−t^5)]_0^1`
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