Section 8.2 – Integration by Parts

Find the Error The following is an example of a student response. How can you tell the final answer is incorrect? Where did the student make an error? The integral of a product is not equal to product of the integrals. This should remind us of the Product Rule. Is there a way to use the Product Rule to investigate the antiderivative of a product?

Integration by Parts: An Explanation When u and v are differentiable functions of x : The Product Rule tells us… If we integrate both sides… If we simplify the integrals… If we solve for one of the integrals…

Integration by Parts Rewrite the function into the product of u and dv. The integral equals… u times the antiderivative of dv. The integral of the product of the antiderivative of dv and the derivative of u.

Integration by Parts: The Process

Example 1 Pick the u and dv. Find du and v. Apply the formula. Differentiate.Integrate.

Example 2 Pick the u and dv. Find du and v. Apply the formula. Differentiate.Integrate. You may need to apply Integration by Parts Again. Pick the u and dv. Find du and v. Apply the formula.

White Board Challenge

Example 3 Since, multiple Integration by Parts are needed, a Tabular Method is a convenient method for organizing repeated Integration by parts. Repeated DifferentiationRepeated Integration + – + – Must get 0. Start with + Alternate Find the sum of the products of each diagonal: Differentiate the u. Integrate the dv. Connect the diagonals. Notice the cubic function will go to zero. So it is a good choice for u.

Example 4 Pick the u and dv. Find du and v. This was a bad choice for u and dv. Differentiate.Integrate.

Example 4: Second Try Pick the u and dv. Find du and v. Apply the formula. Differentiate.Integrate. Try the opposite this time.

Example 5 If there is only one function, rewrite the integral so there is two. Pick the u and dv. Find du and v. Apply the formula. Differentiate.Integrate.

Example 6 Pick the u and dv. Find du and v. Apply the formula. You may need to apply Integration by Parts Again. Apply the formula. Pick the u and dv. Find du and v. If you see the integral you are trying to find, solve for it.

Example 7 If there is only one function, rewrite the integral so there is two. Pick the u and dv. Find du and v. Apply the formula.

Integration by Parts: Helpful Acronym When deciding which product to make u, choose the function whose category occurs earlier in the list below. Then take dv to be the rest of the integrand. L I A T E ogarithmic nverse trigonometric lgebraic rigonometric xponential

White Board Challenge