# Thread: Integration with substitution

1. ## Integration with substitution

Hey I was wondering if I can have some help with an integration problem. The integral is cos(x)/sqrt(x) dx. They tell me to use the substitute u = 2 sqrt(x).

The u is nice because when we find du = 1/sqrt(x) dx. Now im stuck with the integral of cos(x) du.

I tried to solve for x in terms of u and x = (u^2)/4

Replacing I get the integral of cos(u^2/4)du.

This is where im stuck.

Any help is appreciated. Thanks

2. Originally Posted by jlip
Hey I was wondering if I can have some help with an integration problem. The integral is cos(x)/sqrt(x) dx. They tell me to use the substitute u = 2 sqrt(x).

The u is nice because when we find du = 1/sqrt(x) dx. Now im stuck with the integral of cos(x) du.

I tried to solve for x in terms of u and x = (u^2)/4

Replacing I get the integral of cos(u^2/4)du.

This is where im stuck.

Any help is appreciated. Thanks
Are you sure it's not cos(x) sqrt(x) dx ....?

3. yep it is cos(x) divided by the square root of x

4. You can use the identity that converts $\cos^2 x$ into something involving $\cos 2x$ - it should be at your fingertips but unfortunately it's not at mine!

Then you'll have something in the form $\cos 2u$ or something, and you should be able to integrate that.

5. Originally Posted by Matt Westwood
You can use the identity that converts $\cos^2 x$ into something involving $\cos 2x$ - it should be at your fingertips but unfortunately it's not at mine!

Then you'll have something in the form $\cos 2u$ or something, and you should be able to integrate that.
No. If the problem is correct (and I have my doubts that it is), then you have to integrate $\cos (u^2)$. This is not the same as $\cos^2 (u)$. In fact, $\int \cos (u^2) \, du$ cannot be found using a finite sum of elementary functions (which is what I'm assuming the OP wants to do). It's a Fresnel integral: Fresnel integral - Wikipedia, the free encyclopedia

6. Originally Posted by mr fantastic
No. If the problem is correct (and I have my doubts that it is), then you have to integrate $\cos (u^2)$. This is not the same as $\cos^2 (u)$. In fact, $\int \cos (u^2) \, du$ cannot be found using a finite sum of elementary functions (which is what I'm assuming the OP wants to do). It's a Fresnel integral: Fresnel integral - Wikipedia, the free encyclopedia
D'oh! Sorry, goes to show how useful LaTeX is in making things clearer. Reading raw unrendered code is trickier than it looks.