Integration Strategy

Main idea

Derivatives follow rules; integrals take judgment. The trick is to look before you leap: simplify first, check for an easy substitution, and only then match the integrand's shape to one of the techniques from this unit.

Explanation

An integration strategy is simply a plan for deciding which technique to use on an integral you've never seen before. It's good for the real situation you'll face on exams and in engineering, where nobody tells you which section a problem came from. All we're really saying is: before you start grinding, take a second to look at what kind of integral you've got. There's no single formula in this lesson, just a short checklist with a few steps: simplify (clean up the algebra), look for a substitution (is the derivative of one piece sitting right there?), and match the shape (rational function, root, trig powers, and so on) to the technique built for it. Most of the work is recognizing which situation you're in. It sounds like a lot of choices, but it's really just a few questions asked in order; bear with it.

When you learned derivatives, every function came with a rule: product rule, chain rule, done. Integrals aren't like that. The product rule has a reverse (parts), and so does the chain rule (substitution), but neither one tells you when to use it, and lots of integrals need some rewriting before any technique applies. So the skill now is less about computing and more about recognizing.

1. Simplify. Expand, split fractions, cancel, or use an identity. It sounds too obvious to list, but it's the step people skip most. $\displaystyle\int \frac{3}{x^{-1/2}(x^{3/2} - x^{1/2})}dx$ looks scary and is really $\displaystyle\int \frac{3}{x - 1}dx$ .

2. Look for a substitution. Is one part of the integrand (nearly) the derivative of another part? $\int x\sec(x^2)\tan(x^2)\,dx$ wants $u = x^2$ . Substitution is the cheapest technique, so it's worth checking before anything heavier.

3. Classify by shape.

If you see…	Try…
A rational function	Partial fractions (divide first if needed)
$x^n$ times $e^x$ , $\sin x$ , $\cos x$ ; or $\ln x$ , $\arctan x$ alone	Integration by parts
Powers of $\sin$ and $\cos$ , or $\tan$ and $\sec$	Trig integrals
$\sqrt{a^2 - x^2}$ , $\sqrt{a^2 + x^2}$ , $\sqrt{x^2 - a^2}$	Trig substitution
$\sqrt[n]{ax + b}$	Root substitution
A quadratic that won't factor	Complete the square

4. Try again. If nothing fits, rewrite the integrand: multiply by a conjugate, use an identity, split it in two, or combine methods. Many integrals need two techniques in a row, like a substitution that leaves something for parts.

5. Know when to stop. Some integrals have no elementary antiderivative at all: $\int e^{-x^2}dx$ , $\int \frac{\sin x}{x}dx$ , $\int \sqrt{1 + x^3}\,dx$ . That's not because nobody's clever enough; it's been proven that no combination of the usual functions works. For those you approximate (later in this unit) or use a series.

A common mistake. It's tempting to grab the first technique that comes to mind, often parts, and push through pages of algebra. If the integral is getting worse after a step instead of better, that's your signal to back up and try something else. A good choice usually makes the next line visibly simpler.

See it

This is xⁿe⁻ˣ, a power times an exponential, which is a job for integration by parts. Doing parts n times shows the area from 0 to ∞ is n!, and you can check it: at n = 1, 2 and 3 the area reads 1, 2 and 6. Notice the area also changes smoothly for the values in between. That smooth version is the gamma function, and it's how mathematicians made sense of (1/2)! = √π/2.

Worked examples

Example 1

Find $\displaystyle\int x\sec(x^2)\tan(x^2)\,dx$ .

1
Before reaching for anything fancy, notice the $x$ in front is half the derivative of $x^2$ . That's a substitution: $u = x^2$ , $du = 2x\,dx$ .

Example 2

Find $\displaystyle\int x^3\ln x\,dx$ .

1
No substitution jumps out, and it's a power of $x$ times a log, so it's a job for parts with the log as $u$ : $u = \ln x$ , $dv = x^3\,dx$ , so $du = \frac{dx}{x}$ , $v = \frac{x^4}{4}$ .

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Which technique fits $\displaystyle\int \frac{x^2}{\sqrt{9 - x^2}}\,dx$ best?

Problem 2+5 XP

Evaluate $\displaystyle\int_1^e \frac{\ln x}{x}\,dx$ .

Problem 3+5 XP

Find $\displaystyle\int x^2(\ln x)^2\,dx$ .

Problem 4+5 XP

Evaluate $\displaystyle\int_0^{\pi/2} \cos^3 x\sin(2x)\,dx$ .

Problem 5+5 XP

Find $\displaystyle\int x^5 e^{-x^3}\,dx$ .

Problem 6+5 XP

Find $\displaystyle\int \frac{\sqrt{1 + x}}{\sqrt{1 - x}}\,dx$ .

Problem 7+5 XP

Doctors measure a drug's total exposure as the area under its concentration curve. For $C(t) = 8te^{-0.5t}$ mg/L, find $\int_0^\infty C(t)\,dt$ .

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

Find $\displaystyle\int\ln\left(x^2 + 1\right)dx$ .

Where it's used

Computer algebra systems run a rigorous version of this checklist (the Risch algorithm). When it proves no elementary antiderivative exists, they switch to numerical integration, the subject of the next lesson but two.

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