Choosing an integration technique

Warm-up

Three quick questions from earlier lessons.

Discover

$\displaystyle\int x\cos(x^2)\,dx$ and $\displaystyle\int x\cos x\,dx$ look alike. Which one is a substitution?

$\displaystyle\int x\cos(x^2)\,dx$
$\displaystyle\int x\cos x\,dx$

Key idea

Read the integrand before you start.

Substitution: an inside function with its derivative (up to a constant) as a factor. Parts: a product of unrelated types, such as a power times

e^x

,

\sin x

or

\ln x

. Partial fractions: a rational function whose denominator factors. Algebra first: divide, or complete the square.

Worked example

Choose a method for three integrals.

\displaystyle\int 2x(x^2 + 1)^5\,dx

: substitution

x² + 1 inside, 2x outside.

\displaystyle\int x e^{-x}\,dx

: parts

Power times exponential.

\displaystyle\int \dfrac{dx}{x^2 - 4}

: partial fractions

x² − 4 = (x − 2)(x + 2).

Your turn, guided

Your turn: $\displaystyle\int \dfrac{2x}{x^2 + 9}\,dx$ is a substitution with $u = x^2 + 9$ .

\ln(x^2 + 9) + C

∫du/u.

Practice ★

Which method suits $\displaystyle\int x e^{ 2x^2}\,dx$ best?
1. Substitution, $u = 2x^2$
2. Integration by parts
3. Partial fractions

Which method suits $\displaystyle\int x e^{ 6x}\,dx$ best?
1. Integration by parts, $u = x$
2. Substitution, $u = 6x$
3. Partial fractions

Watch out!

Practice ★★

Which method suits $\displaystyle\int \dfrac{dx}{x^2 - 36}$ best?
1. Partial fractions: $x^2 - 36 = (x - 6)(x + 6)$
2. The arctangent formula
3. Integration by parts

Missing exercise F.INT.21-t2

Exit ticket

Three questions to check you have got it.

Which method suits $\displaystyle\int x e^{ 6x}\,dx$ best?
1. Integration by parts, $u = x$
2. Substitution, $u = 6x$
3. Partial fractions
Which method suits $\displaystyle\int x e^{ 3x}\,dx$ best?
1. Integration by parts, $u = x$
2. Substitution, $u = 3x$
3. Partial fractions
Which method suits $\displaystyle\int x e^{ 4x}\,dx$ best?
1. Integration by parts, $u = x$
2. Substitution, $u = 4x$
3. Partial fractions

Summary card

Which technique?

Inside + its derivative: substitution.
Unrelated product: parts.
Factorable denominator: partial fractions.

Watch out: $x^2 + a^2$ (arctan) and $x^2 - a^2$ (fractions) differ.

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