Integration by Parts
Reverse the product rule to trade one integral for another that is easier, and choose the parts so the trade is an improvement.
What a learner can do afterwards
- Choose u and dv so the new integral is simpler than the original
- Apply the method twice and solve for the original integral when it reappears
- Integrate ln x by taking dv = dx
1 · Read
Some products cannot be split by substitution, like x sin x. Read the product rule backwards instead: the integral of u dv equals u v minus the integral of v du. You trade one integral for another, so choose the trade that gets simpler.
Take the integral of x e^x dx. LIATE puts algebra before exponentials, so set u = x and dv = e^x dx. Then du = dx and v = e^x. You get x e^x minus the integral of e^x dx, which is x e^x - e^x + C. The reverse split would give a harder integral, so the choice matters.
A lone log is welcome too: for ln x, set u = ln x and dv = dx. Sometimes the old integral returns, as with e^x cos x, and then you collect it on the left and solve for it. A square like x^2 e^x simply needs two full rounds.
Pick u by LIATE: logs, then inverse trig, then algebra, with trig and exponentials as dv. Whatever you call dv must be something you can integrate. Check your answer by differentiating, since a sign slip on the minus term is the classic error.
Split into u and dv with LIATE, subtract the new integral, and solve when the old one returns.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.