What is wrong with my integral? $\sin^5 x\cos^3 x$

Question

I am trying to do an integration problem but am running into a problem! My answer is different from what the solution says. My attempt:

Evaluate $$\int\cos^3x\sin^5 x\mathop{dx}$$

$\int\cos^3x\sin^5 x\mathop{dx}=(1-\cos^2 x)^2\sin x \cos ^3 x$

Then with u substitution, letting $u=\cos x \implies du=-\sin x \mathop{dx}$

which gives us

$-\int (1-u^2)^2\cdot u^3\mathop{du}=\int-u^3+2u^5+u^7=-(\cos^ 4 x)/4+(\cos^6 x)/3-(\cos ^8 x)/8 +C$

However the solution books says the answer is

$(\sin^6 x)/6-(\sin ^8 x)/8 +C$

Answer 1

They are equivalent, with different constants of integration; specifically, they differ by $1/24$. The proof of this I leave to you as an exercise, but a simple way to do it is to convert all the cosines in your expression to sines, via the circular identity $\sin^2 x + \cos^2 = 1$.

To figure out how the book got a different answer, note that instead of converting $\sin^4 x$ in terms of cosine, you could have also converted $\cos^3 x = (1-\sin^2 x)\cos x$, which leads to a simpler integrand--the one the book obtained.

Answer 2

Let's see a simpler case: $$ \int 2\sin x\cos x\,dx $$ You have two choices: either do $u=\sin x$, so $\cos x\,dx=du$, and you get $$ \int 2u\,du=u^2+c=\sin^2x+c $$ or do $v=\cos x$, so $\sin x\,dx=-dv$, and you get $$ \int -2v\,dv=-v^2+c=-\cos^2x+c $$ Which one is right? Both, of course, but this doesn't mean you reached the false conclusion that $\sin^2x=-\cos^2x$.

The fact is that an antiderivative is only determined up to a constant, so what you can say is that $$ \sin^2x=-\cos^2x + k $$ for some constant $k$; you surely know that $k=1$, in this case.

Your problem is exactly the same. You happened to use the second substitution instead of the first one. Try and determine $k$ such that $$ -(\cos^ 4 x)/4+(\cos^6 x)/3-(\cos ^8 x)/8=(\sin^6 x)/6-(\sin ^8 x)/8 +k $$

Hint: evaluate at $0$ both expressions.