MHB How Do You Integrate Trigonometric Functions with Substitution?

[annie122]

how do i integrate sin(pi*x)*sqrt(1 + pi*2*cos(pi*x)^2)?
i reduced this to sqrt(u^2-u) but i don't know how to go from here

 

[MarkFL]

Re: integration

I think what you have written should be interpreted as:

$$\int \sin(\pi x)\sqrt{1+2\pi\cos^2(\pi x)}\,dx$$

Is this correct? And if so, what substitution did you use?

 

[Prove It]

Re: integration

how do i integrate sin(pi*x)*sqrt(1 + pi*2*cos(pi*x)^2)?
i reduced this to sqrt(u^2-u) but i don't know how to go from here

Substitute \displaystyle \begin{align*} u = \cos{ \left( \pi \, x \right) } \implies du = -\pi\sin{ \left( \pi \, x \right) } \, dx \end{align*} to start with...