Integration by parts is a technique used to integrate the product of two functions. It is based on the product rule for differentiation and is given by the formula: ∫u dv = uv - ∫v du, where u and dv are differentiable functions of x.
Choosing u and dv can be tricky, but a common heuristic is the LIATE rule, which stands for Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential functions. Generally, you choose u to be the function that comes first in this list and dv to be the remaining part of the integrand.
If applying integration by parts once does not simplify the integral sufficiently, you may need to apply it multiple times. After each application, you should simplify the resulting integrals and check if further applications are necessary.
Yes, integration by parts can be used for definite integrals. The formula is modified to account for the limits of integration: ∫[a,b] u dv = [uv] from a to b - ∫[a,b] v du. You compute uv at the bounds and then subtract the integral of v du over the same interval.
If the integral cycles back to the original integral after applying integration by parts, you can solve for the original integral algebraically. Set up an equation where the original integral appears on both sides, then solve for the integral by isolating it on one side of the equation.