Integration by Parts: Solving an Intricate Integral

[Panphobia]

Homework Statement

∫x*cos(x^2) dx

I tried using integration by parts, but the integral of cos(x^2) is very long, and I couldn't get it completely with my knowledge at the moment, so is there an easier way to solve this problem?

 

[Dick]

Homework Statement

∫x*cos(x^2) dx

I tried using integration by parts, but the integral of cos(x^2) is very long, and I couldn't get it completely with my knowledge at the moment, so is there an easier way to solve this problem?

Yes, don't use integration by parts. Use u substitution. Put u=x^2.

 

[Panphobia]

So the definite integral [0, sqrt(pi)] would be 0 correct?

 

[Mark44]

Yes, correct.
Also, when you're working with integrals, it's usually best to see if a simple substitution will work before tackling it with integration by parts. Integration by substitution is usually a simpler approach that integration by parts, so if it doesn't work out, you haven't wasted much time.

In this case, and as you saw, it's a very obvious substitution that works.

BTW, when you post a problem, you need to show what you have tried, even if it wasn't successful. That's a rule in this forum.