I am looking at a problem integrating sinh^5(x)cosh(x) dx. They set u=sinh so then du=cosh, then they get a final answer of sinh^6x/6. I am not understanding what happened to the coshx can some one please explain this to me?
Thank you!!
I am looking at a problem integrating sinh^5(x)cosh(x) dx. They set u=sinh so then du=cosh, then they get a final answer of sinh^6x/6. I am not understanding what happened to the coshx can some one please explain this to me?
Thank you!!
first off, notice that when you differentiate your answer you get the original. where did cosh(x) go? it was replaced when we made the change of variable. replaced with what? well, let's see...
$\displaystyle \int \sinh^5 x \cosh x ~dx$
Let $\displaystyle u = \sinh x$
$\displaystyle \Rightarrow du = \cosh x ~dx$
that is what happened to the cosh(x), it was replaced by 1 when we changed the variable to u. so cosh(x)dx became du
So our integral becomes:
$\displaystyle \int u^5 ~du$
the rest follows.
taking the derivative of the answer by the chain rule gets us back to the original integrand, which should happen
You're forgetting one of the basic rules of integration, me thinks. There's a general rule when it comes to integrating:
$\displaystyle f(x) [f(x)]^n $ w.r.t. x.
It equals $\displaystyle \frac{[f(x)]^{n+1} }{n+1} $
If you're still not convinced, what do you get when you differentiate $\displaystyle \frac{ \sin^6 (x)}{6} $?