Techniques of integration

Many integrals involve a product of two functions, such as a polynomial times an exponential or a trigonometric factor. The key tool is integration by parts, which reverses the product rule for differentiation. Write the integrand as $udv$, choose $u$ to be the part that simplifies when differentiated, and $dv$ to be the part that can be integrated easily. Then $\int udv=uv-\int vdu$. This single idea lets you turn a difficult integral into a simpler one that you can finish directly or with another round of the same technique.