Integration by Parts
This technique is used when the integrand is a product of two functions. It’s based on the product rule for differentiation.
Formula: $ \int u dv = uv - \int v du $
Choosing u and dv: The choice of $ u $ and $ dv $ is crucial and often involves using the “LIATE” rule (Logarithmic, Inverse Trigonometric, Algebraic, Trigonometric, Exponential). Prioritize choosing $ u $ according to this order.
Example: $ \int x e^x dx $
Let $ u = x $ , $ dv = e^x dx $ . Then $ du = dx $ , $ v = e^x $ .
$ \int x e^x dx = xe^x - \int e^x dx = xe^x - e^x + C $