# Thread: Integral by substitution with trig functions

1. ## Integral by substitution with trig functions

I'm trying to evaluate the integral of sinxcos(cosx)dx. Would I want to use sinx as my u and then du would be cos x. and So I'd be left with cos x times the integral of u?

2. That's not the proper substitution.

If you put $u=\cos x$ then $-du=\sin x\,dx.$ You actually have the $\sin x\,dx$ and the $\cos x$ is replaced by $u.$ Do the substitutions.

3. Originally Posted by Krizalid
That's not the proper substitution.

If you put $u=\cos x$ then $-du=\sin x\,dx.$ You actually have the $\sin x\,dx$ and the $\cos x$ is replaced by $u.$ Do the substitutions.
Alright, but is it possible to do it the other way? Because it would still lead to a term canceling out, or is there only one way to solve problems like this? How would I know to use sinx as u rather than cos x?

4. Just in case a picture helps...

You have to choose it a particular way round in order to work backwards through the chain-rule for differentiation.

You're trying to work out how the given expression might have resulted from differentiating according to this rule, so that it fits the pattern

$f'(g(x))\ g'(x)$

or, in other words, fits into the bottom level of the picture. Then we can integrate up the straight dashed line with respect to the dashed balloon expression. Imagine the dashed balloon is empty, or indeed is just a variable, and you should be able to find the function F...

Don't integrate - balloontegrate!

Balloon Calculus: worked examples from past papers

5. Originally Posted by Krizalid
That's not the proper substitution.

If you put $u=\cos x$ then $-du=\sin x\,dx.$ You actually have the $\sin x\,dx$ and the $\cos x$ is replaced by $u.$ Do the substitutions.
So would I just be left with negative one times the integral of cos(u) as the next step

Just to check would my final answer of -sin(cosx)+C be correct?

6. Originally Posted by fattydq
Just to check would my final answer of -sin(cosx)+C be correct?
If you are doing integration, you must have done differntiation