Evaluate the integral of cos(x)sin(x)(1+ sin(x))^3 with respect to x.
Substitution of 'u=1+sin(x)' is required.Differentiating this with respect to x gives cos(x)... therefore du=1/cos(x) dxmultiplying that through leaves the integral of sin(x)(1+sin(x))^3 which therefore can be replaced using the substitution of u=1+sin(x) to give the integral of (u-1)u^3 with respect to 'u' since sin(x) is equal to 'u-1.'Expanding this gives the simple integral of u^4 - u^3.Evaluating gives u^5/5 - u^4/4 +C.Replacing back with the initial substitution gives the answer as (1 +sin(x))^5/5 - (1+sin(x))^4/4 +C
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