When should I use integration by parts and how do I pick u?

Integration by parts uses the formula integral of u·dv = u·v − integral of v·du. The trick is choosing u so that du is simpler and dv is easy to integrate. Use the ILATE order to pick u: Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential. Whichever type comes first in ILATE becomes u, the rest is dv. For example in x·e^x dx, x is Algebraic and e^x is Exponential, so u = x. Then du = dx and dv = e^x dx gives v = e^x, leading to x·e^x − integral of e^x dx = x·e^x − e^x + C. If your integral got harder, you almost certainly picked u and dv the wrong way around.