AP Calculus AB Integration by Substitution Lesson (Free PDF)

AP Calculus AB math lesson: Integration by Substitution – Maths PDF

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This u substitution lesson for AP Calculus AB covers integration by substitution (Common Core AP CALC AB 6.9, AP CALC AB 6.10, AP CALC AB 6.11, AP CALC AB 6.12, AP CALC AB 6.14), with a free PDF to print.

Integration by substitution reverses the chain rule. Choose u as the inner function whose derivative appears in the integrand, replace u and du, integrate in u, and substitute back. For definite integrals, convert the limits to u. When substitution alone is not enough, use long division or complete the square first.

AP Calculus AB · Common Core State Standards: AP CALC AB 6.9, AP CALC AB 6.10, AP CALC AB 6.11, AP CALC AB 6.12, AP CALC AB 6.14

Key facts in this u substitution lesson

Key facts

Let u = g(x) and du = g′(x) dx, then the integral becomes the integral of f(u) du.
Adjust constant factors: if du is off by a constant, divide or multiply to compensate.
For definite integrals, change the limits to u-values or return to x before evaluating.
Use long division for improper rational functions and completing the square for irreducible quadratics.

What is integration by substitution?

Substitution reverses the chain rule. If an integrand has the form \(f(g(x))\,g^{\prime}(x)\), set \(u=g(x)\) so that \(du=g^{\prime}(x)\,dx\), and the integral becomes the simpler \(\int f(u)\,du\).

A five-step routine for integration by substitution.

Example. Find \(\int 2x(x^{2}+1)^{3}\,dx\). Let \(u=x^{2}+1\), so \(du=2x\,dx\). Then \(\int u^{3}\,du=\frac{u^{4}}{4}+C=\frac{(x^{2}+1)^{4}}{4}+C\). Check by differentiating with the chain rule.

How do you choose u?

Look for an expression whose derivative (up to a constant factor) also appears in the integrand. Good candidates are the inside of a power, the exponent of \(e\), the argument of a trig function, or the denominator.

Patterns that signal a substitution.

If the derivative is off by a constant, fix it: for \(\int x\,e^{x^{2}}dx\), \(du=2x\,dx\) gives \(x\,dx=\frac{1}{2}du\), so the answer is \(\frac{1}{2}e^{x^{2}}+C\).

How do definite integrals change?

Either convert the limits to \(u\) and never go back to \(x\), or find the antiderivative in \(x\) and use the original limits. Do not mix the two.

When you substitute in a definite integral, convert the limits to u.

\(\int_{0}^{2}2x(x^{2}+1)^{3}dx=\int_{1}^{5}u^{3}du=\left[\frac{u^{4}}{4}\right]_{1}^{5}=\frac{625-1}{4}=156\).

The shaded area equals ∫₀¹ 2x e^(x²) dx = e − 1 ≈ 1.718.

Check your answer by differentiating. If the chain rule returns the original integrand, the antiderivative is right.
Keep the constant \(+C\) in every indefinite integral.
Never leave \(x\) and \(u\) mixed in the same integral. Everything, including \(dx\), must be written in \(u\) before you integrate.

For example, in \(\int x\sqrt{x^{2}+4}\,dx\) with \(u=x^{2}+4\), the factor \(x\,dx\) is replaced by \(\frac{1}{2}du\), giving \(\frac{1}{2}\int u^{1/2}du=\frac{1}{3}u^{3/2}\). If a stray \(x\) remains after the substitution, either another choice of \(u\) is needed or the integral requires a different method.

What if substitution is not enough?

For a rational function whose numerator has degree at least the denominator, use long division first: \(\frac{x^{2}+3x+1}{x+1}=x+2-\frac{1}{x+1}\). For \(\frac{1}{x^{2}+4x+13}\), complete the square to get \(\frac{1}{(x+2)^{2}+9}\), and then use \(\int\frac{1}{u^{2}+a^{2}}du=\frac{1}{a}\tan^{-1}\frac{u}{a}\).

U substitution lesson: questions and answers

What is u-substitution?

It is a technique that rewrites an integral in terms of a new variable u = g(x) so that the integrand becomes simpler. It undoes the chain rule.

How do you choose u?

Pick the inner function (inside parentheses, a root, an exponent or a denominator) whose derivative, up to a constant, also appears in the integrand.

What happens to the limits of a definite integral?

If you rewrite in u, convert each limit with u = g(x). Then evaluate in u without returning to x. Alternatively, find the antiderivative in x and use the original limits.

What if du does not match exactly?

A constant factor can be adjusted. For example, if du = 2x dx but the integral has x dx, replace x dx by (1/2) du.

When do you use long division in integration?

Use it when the numerator of a rational function has degree greater than or equal to the denominator. Divide first, then integrate each term.

How to use this u substitution lesson

Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.

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