Strategies of Integration

After $u$ -substitution, integration by parts is by far the most important technique to learn. It converts a hard integral $\int u\,dv$ into an easier integral $uv - \int v\,du$ . The tricky thing is figuring out what to pick for $u$ and $dv$ . We want to choose $u$ so that $du$ is simpler than $u$ , and $dv$ so that $v$ isn't much more complicated than $dv$ . Remember that:

Some functions get a lot simpler when differentiated or a lot more complicated when integrated. These almost always want to be part of $u$ .

Some functions get a little simpler when differentiated. These are a lower priority to include in $u$ .

Some functions, like $e^x$ and $\sin(x)$ , don't get simpler at all. These have still lower priority, and usually are part of $dv$ .

Some functions actually get simpler when integrated and are almost always part of $dv$ .

If we can pick $u$ from higher on the list and $dv$ from lower on the list, then $\int v\,du$ will be simpler than $\int u\,dv$ .

If we have two terms at the same level, then we can try using some algebra to relate $\int v\,du$ to $\int u\,dv$ . We may have to integrate by parts twice to make this work.

Example: Compute

$\int\frac{\ln(x)}{x^3}\,dx.$

Solution: Since

$\displaystyle\Bigl(\ln(x)\Bigr)'=\frac{1}{x}$ is simpler than

$\ln(x)$ and

$\displaystyle\int x^{-3}\, dx$ is simpler than

$x^{-3}$ , we take

$u=\ln(x)$ and

$\displaystyle dv=\frac{1}{x^3}\,dx$ . Then

$du=\frac{1}{x}, \quad \text{ and }\quad v=-\frac{1}{2x^2}.$ Hence

$\int\frac{\ln(x)}{x^3}\,dx=-\frac{\ln(x)}{2x^2}-\left(\int -\frac{1}{2x^3}\,dx\right)=-\frac{\ln(x)}{2x^2}-\frac{1}{4x^2}+C.$

The Fundamental Theorem of Calculus

The Indefinite Integral and the Net Change

Substitution

Area Between Curves

Volumes

Integration by Parts

Integrals of Trig Functions

Trig Substitutions

Partial Fractions

Strategies of Integration

Improper Integrals

Differential Equations

Models of Growth

Infinite Sequences

Infinite Series

Integral Test

Comparison Tests

Convergence of Series with Negative Terms

The Ratio and Root Tests

Strategies for testing Series

Power Series

Representing Functions as Power Series

Taylor and Maclaurin Series

Applications of Taylor Polynomials

Partial Derivatives

Multiple Integrals

Iterated Integrals over Rectangles

Double Integrals over General Regions

Integration by Parts