6.11 Integrating Using Integration by Parts

Learning objectives

—FUN-6.E—For integrands requiring integration by parts: (a) Determine indefinite integrals. (b) Evaluate definite integrals• FUN-6.E For integrands requiring integration by parts: (a) Determine indefinite integrals. (b) Evaluate definite integrals.• FUN-6.E.1 Integration by parts is a technique for finding antiderivatives.• Enduring understanding FUN-6: Recognizing opportunities to apply knowledge of geometry and mathematical rules can simplify integration.

The product rule gives $(uv)'=u'v+uv'$ . Integrating and rearranging produces integration by parts: choose one factor as $u$ to differentiate and the remaining factor with $dx$ as $dv$ to integrate.

\int u,dv=uv-\int v,du

Step	Decision	Check
1	choose $u$	differentiating it should simplify it
2	choose $dv$	it must have an antiderivative you can find
3	find $du$ and $v$	include $dx$ consistently
4	substitute into the rule	the new integral should be simpler than the original

For $\int xe^x\,dx$ , take $u=x$ and $dv=e^x\,dx$ . Then $du=dx$ and $v=e^x$ , so $\int xe^x\,dx=xe^x-\int e^x\,dx=xe^x-e^x+C$ . Differentiating the result returns $xe^x$ .

For a definite integral, keep the bounds: $\int_0^1 xe^x\,dx=[xe^x]_0^1-\int_0^1e^x\,dx=[xe^x-e^x]_0^1=0-(-1)=1$ . The result is a number, so no $+C$ is added.

Do not choose $u$ by a memorized ordering alone: the real test is whether $u$ is easier after differentiation and $dv$ is easy to integrate. For definite integrals, either retain the original bounds throughout or first find an antiderivative and then evaluate both endpoints—do not mix the two methods or drop a boundary term.

Next Topic

—6.12 Integrating Using Linear Partial FractionsRead