Why Does the Integral of 2xCos(Pi*x^2)dx Not Simplify Using Standard Methods?

[Chaz706]

The General integral for a trig form works whenever the variable inside goes to the first degree.
Example: Sin(x)

But the general integral form for when the variable inside goes beyond the first degree doesn't work.
Example: Sin(x^2), Cos(x^3)

I end up getting an integral whose derivative isn't the original function that I integrated. According to the Fundamental Theory of Calculus, these algorithms can't be correct in these cases.

So how should I solve the integral of 2xCos(Pi*x^2)dx ? I've thought about two things: Substitution and Integration by Parts. Substitution could work, but I get hung up on how to get du. Parts I've tried, but I'm hung up on how to integrate that ugly cosine. Is there another method? Does substitution work? If it does, what's the du? Does Parts work? and how would it work if it does?

Reason why I'm asking: this is one large assignment, and my brain's in knots already from the rest of it.
Furthermore: this is my first post. How do you get that cool coded stuff that makes your integrals look like... integrals?

 

[Jameson]

Ok, I think we'll do substitution.

let $$u = \pi{x^2}$$
$$du = 2\pi{x}dx$$

$$\frac{du}{\pi} = 2xdx$$

Now make the substitution:

$$\int 2x\cos{\pi{x^2}}dx = \frac{1}{\pi}\int\cos{u}du$$

I think you can take it from here.

Jameson

 

[OlderDan]

How do you get that cool coded stuff that makes your integrals look like... integrals?

https://www.physicsforums.com/showthread.php?t=8997

 

[Chaz706]

Thanks

Thanks for your help Jameson. And Older Dan too.