Splitting the Product Rule
Integration by parts trades one integral for another, and choosing well makes the trade a bargain.
Where It Comes From
The product rule says the derivative of u times v is u prime v plus u v prime. Integrate both sides, rearrange, and the left over is the parts formula. It exists for products that u-substitution cannot touch, where two unrelated factors are multiplied together.
\int u\,dv = uv - \int v\,duChoosing u and dv
Pick u to be the factor that simplifies when differentiated, and dv to be the piece you already know how to integrate. Logarithms and inverse functions make good u; exponentials and pure powers make good dv. A bad choice makes the new integral harder than the old one, which is the signal to swap.
u = \ln x,\quad dv = x\,dxA Worked Trade
Consider the integral of x times e to the x. Let u = x so du = dx, and let dv = e to the x dx so v = e to the x. The formula gives x e to the x minus the integral of e to the x, which is x e to the x minus e to the x plus C. The new integral was easier, exactly what a good choice guarantees.
\int x e^x\,dx = xe^x - e^x + CFlashcard
The integration by parts formula
Check your understanding
When integrating x cosine of 2x by parts, which assignment of u and dv leads to a simpler integral?