∫(sinx)^3(cosx)^3dx different answers depending on U-sub?

[LearninDaMath]

∫

Homework Statement

I expanded (sinx)^3 into ∫[(sinx)^2(sinx)(cosx)^3]dx then to ∫[(1-cosx^2)(sinx)(cosx)^3]dx

so then u = sinx

However the official solution for this problem expands (cosx)^3 to get ∫[(cosx)^2(cos)(sinx)^3]dx then to ∫[(1-sinx^2)(cosx)(sinx)^3]dx

so then u = cosx

So the final answer is almost the same for each method except for the fact that the first answer is in terms of sinx and the second final answer is in terms of cosx (and contains an extra negative sign due to derivative of cosx producing a negative).

So the question is, are these two answers equal even though they consist of completely different functions in the final answer?

 

[SammyS]

∫

Homework Statement

I expanded (sinx)^3 into ∫[(sinx)^2(sinx)(cosx)^3]dx then to ∫[(1-cosx^2)(sinx)(cosx)^3]dx

so then u = sinx

However the official solution for this problem expands (cosx)^3 to get ∫[(cosx)^2(cos)(sinx)^3]dx then to ∫[(1-sinx^2)(cosx)(sinx)^3]dx

so then u = cosx

So the final answer is almost the same for each method except for the fact that the first answer is in terms of sinx and the second final answer is in terms of cosx (and contains an extra negative sign due to derivative of cosx producing a negative).

So the question is, are these two answers equal even though they consist of completely different functions in the final answer?

Subtract one answer from the other.

They only differ by a constant.

 

[LearninDaMath]

Ah interesting, thanks.