Integration by Parts is a technique used to integrate the product of two functions. Use it when you have an integral that can be expressed in the form ∫u dv, where you can easily find v and ∫v du.
Use the acronym LIATE (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) as a guide. The function that comes earlier in the list is usually a good choice for u.
The integral might become more complicated, or you might end up back where you started. If this happens, try switching your choices for u and dv.
The formula is ∫u dv = uv - ∫v du. Remember to apply this formula carefully, ensuring you correctly identify u, dv, du, and v.
Evaluate uv at the upper and lower limits of integration, and also evaluate the resulting integral ∫v du within those same limits. Remember to substitute the limits correctly.
Sometimes you need to apply Integration by Parts more than once to solve an integral. Be patient and keep track of your u and dv at each step.
Sometimes, after applying Integration by Parts, you get an integral on the right side that looks similar to the original. In such cases, you can often solve for the original integral algebraically.
Remember to include any constants that arise from integrating dv to find v. These constants can affect the final answer.
Products of polynomials with trigonometric functions (like x*sin(x)), exponentials (like x*e^x), or logarithms (like x*ln(x)) are good candidates for Integration by Parts.
Differentiate your result. If the derivative matches the original integrand, your answer is likely correct.