Remember, when using the substitution rule you are reversing things of the form f(g(x))g'(x). So let u = sin x, then du = cos x. Now, you have to integrate u^6 du, which is 1/7 u^7 du or 1/7 (sin x)^7 . Now do you see what I meant by f(g(x))g'(x), we are basically undoing a derivative obtained using the chain rule.