My textbook says I should solve the following integral by first making a substitution, and then using integration by parts:
$$\int cos\sqrt x \ dx$$
The problem is, after staring at it for a while I'm still not sure what substitution I should make, and hence I'm stuck at the first step. I thought about doing something with the $\sqrt x$, but that doesn't seem to lead anywhere as far as I can tell. Same with the $cos$. Any hints?
Best Answer
make a subs $u = \sqrt x, x = u^2, dx = 2u du$ now the integral $\int \cos \sqrt x \, dx$ is transformed into $$2\int u \cos u \, du = 2 \int u d (\sin u) =2\left( u\sin u - \int \sin u \, du\right) = 2\left( u\sin u +\cos u +C\right)$$