Selecting and checking a BC integration method

Understand the idea

Method selection starts with the structure of the integrand and the conditions of the problem. There is no single integration rule that solves every expression. In BC, algebraic rewriting and substitution sit alongside integration by parts, linear partial fractions, and limits for improper integrals.

Look first for a derivative pattern inside a composition. If the numerator is a multiple of the denominator's derivative, a logarithmic substitution may finish the problem immediately. For a product involving a polynomial and a logarithm or exponential, consider whether integration by parts simplifies one factor.

Factor rational denominators and compare degrees before using partial fractions. Inspect the interval before applying a definite-integral evaluation: a pole or an infinite bound requires a limit. If an exact elementary antiderivative is unavailable, a numerical estimate may be appropriate when the task allows technology.

\text{composition}\to u\text{-substitution};\quad \text{product}\to\text{parts};\quad \text{rational factors}\to\text{partial fractions}

A method you can apply

Simplify the expression and inspect the interval and domain.
Match its structure to a technique and explain what becomes simpler.
Finish the computation, check conditions, and verify an indefinite answer by differentiating.

Worked example

Choose a method and find $\displaystyle \int x\ln x\,dx$ for $\displaystyle x>0$ .

The integrand is a product. Choosing $\displaystyle u=\ln x$ makes $\displaystyle du=dx/x$ , while $\displaystyle dv=x\,dx$ gives $\displaystyle v=x^2/2$ .
Apply integration by parts: $\displaystyle (x^2/2)\ln x-\int(x^2/2)(1/x)\,dx$ .
The remaining integral is $\displaystyle \tfrac12\int x\,dx=x^2/4$ .

The antiderivative is

\displaystyle \tfrac12x^2\ln x-\tfrac14x^2+C

. Differentiating confirms the result on

\displaystyle x>0

.

Try it yourself

Choose an efficient method and find $\displaystyle \int \frac{2x}{x^2+5}\,dx$ .

Show solution and reasoning

Set $\displaystyle u=x^2+5$ , so $\displaystyle du=2x\,dx$ . The integral becomes $\displaystyle \int du/u=\ln(x^2+5)+C$ . The argument is always positive, and substitution is sufficient; partial fractions are unnecessary.