integration by parts

Integration by parts is the technique for integrating a product of two things that have nothing in common — like x times e^x, or x times sin(x) — where no clean renaming helps. Instead of attacking the whole product, you trade it for a different, usually easier, integral. It is a bargain: you give up the hard integral in exchange for an easy boundary term plus a friendlier integral to finish.

It is the product rule run in reverse. The product rule says the derivative of u*v is u'v + uv'; rearranging and integrating gives the formula integral u dv = uv - integral v du. You split the integrand into a part to call u (which you will differentiate) and a part to call dv (which you will integrate), then plug into the formula. For integral x e^x dx, choose u = x and dv = e^x dx; then du = dx and v = e^x, so the result is x e^x - integral e^x dx = x e^x - e^x + C.

The whole art is choosing u and dv well: you want the new integral, integral v du, to be simpler than the one you started with. A common rule of thumb is LIATE — prefer to call u whichever factor comes first in the order Logarithmic, Inverse-trig, Algebraic, Trigonometric, Exponential — because that tends to be the factor that simplifies when differentiated. A bad split just hands you a harder integral, so if your new integral looks worse, swap your choices and try again.