Integration by Parts: Formulas, Strategies, and Applications

All flashcards

Indefinite integration-by-parts formula

$\int u\,dv=uv-\int v\,du$; here $du=u'(x)\,dx$ and $v$ is an antiderivative of $dv$.

Definite integration-by-parts formula

$\int_a^b u(x)v'(x)\,dx=[uv]_a^b-\int_a^b u'(x)v(x)\,dx$, where $[uv]_a^b=u(b)v(b)-u(a)v(a)$.

LIATE

A rule of thumb for choosing $u$: logarithmic, inverse trigonometric, algebraic, trigonometric, then exponential.

Cyclic integral

An integral that returns after repeated integration by parts and can then be isolated algebraically.

Multivariable integration by parts

For a scalar $u$ and vector field $V$, $\int_\Omega u(\nabla\cdot V)\,d\Omega=\int_{\partial\Omega}uV\cdot\hat n\,d\Gamma-\int_\Omega\nabla u\cdot V\,d\Omega$.

Quiz questions

1. Which differentiation rule gives rise to the integration-by-parts formula?Answer: The product rule, $(uv)'=u'v+uv'$.
2. How does LIATE order the categories used as a guide for choosing $u$?Answer: Logarithmic, inverse trigonometric, algebraic, trigonometric, exponential.
3. If two integrations by parts give a cyclic integral equation $I=A-I$, how can $I$ be isolated?Answer: Add $I$ to both sides, then divide by 2.
4. Using $Ff(\xi)=\int e^{-2\pi iy\xi}f(y)\,dy$, what is $Ff'(\xi)$ when the boundary term vanishes?Answer: $2\pi i\xi Ff(\xi)$
5. For $\int x\cos x\,dx$, choose $u=x$ and $dv=\cos x\,dx$. What are $du$ and $v$?Answer: $du=dx$ and $v=\sin x$.