How to integrate cos√x using substitution and integration by parts

[EV33]

Integral of cos√x



We are supposed to use substitution and integration by parts but I really don't know where to even start.



No matter what I substitute for U I will be left without a du.

 

[gabbagabbahey]

what are your limits of integration?...what does your integral become when you use the substitution u=sqrt(x)? What is du?

 

[EV33]

There are no limits it is indefinite. If you let U= √x then du= 1/2√x and that is my problem because I am left with with integral of cos(u) and no du.

 

[gabbagabbahey]

doesn't $du=\frac{1}{2\sqrt{x}}dx$ and doesn't that mean that $dx=2 \sqrt{x} du= 2udu$?

 

[EV33]

Yes it does. So does that mean when I integrate I get u^2 sinu and from there I just need to back substitute?

 

[gabbagabbahey]

Well, that means that

$\int cos(\sqrt{x})dx= \int 2ucos(u)du$

now you'll need to integrate by-parts...try using f(u)=2u and g'(u)=cos(u)du

 

[EV33]

oh ok. Thank you for the help.