What is the integral of $int sin^3(x) cos^2(x) dx$ ?

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Answer 1 · 2016-05-03T10:55:11

$(cos^5 x )/5 - (cos^3 )/3 +C$

Write $int sin^3 cos^2 x dx= int sinx (1-cos^2 x) cos^2 x dx$

Now let cos x =t, so that sin dx = -dt. On this substitution, the integral becomes

$-int (1-t^2)(t^2)dt$

$int (t^4-t^2 )dt$ = $t^5 /5 -t^3 /3 +C$

No back substitution would have $(cos^5 x )/5 - (cos^3 )/3 +C$

What is the integral of int sin^3(x) cos^2(x) dxint sin^3(x) cos^2(x) dx?