How to integrate x*cos(pi*x)dx?

You use integration by parts, which is based on the form:

⌠v du = uv - ⌠u dv

which is just the product rule for differentiation, integrated

In this case, choose v = x, du = cos(πx) dx

Then dv = dx and u = sin(πx)/π

So:

⌠x cos(πx) dx = x sin(πx) / π - 1/π⌠sin(πx) dx = x sin(πx) / π + 1/π^2 cos(πx) + C

or, bringing everything over a common denominator, (πx sin(πx) + cos(πx)) / π^2 + C

You can the answer at Wolfram Alpha, go to this web page:

www.wolframalpha.com

and type in "integrate x cos(pi x)"

integrate x*cos(pi*x)dx

= (x/pi) sin(xpi) + (1/pi^2) cos(xpi) + C

by parts...