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\( \int u\,du \)

Integrating Using
Substitution

Master the technique of \( u \)-substitution to reverse the chain rule and evaluate integrals that resist basic antidifferentiation.

AP Calculus AB

Why Do We Need Substitution?

Many integrals don't match a basic formula directly. Consider \( \int 2x \cos(x^2)\, dx \). There is no standard rule that handles this composite structure. The integrand is a composition of functions multiplied by a derivative: the hallmark of the chain rule in reverse.

The Chain Rule Connection

Recall the chain rule for differentiation:

Integrating both sides reverses this process:

Substitution formalises this reversal by letting \( u = g(x) \) so that \( du = g'(x)\, dx \).

What Is \( u \)?

\( u \)
The inner function in a composition. You choose \( u = g(x) \) to simplify the integral.
\( du \)
The differential of \( u \): \( du = g'(x)\, dx \). It must appear (or be adjustable) in the integrand.

The Substitution Rule

For definite integrals, change the limits too:

Substitution Visualised

Original Integral
\( \int f(g(x))\,g'(x)\,dx \)
\( u = g(x) \)
 Simplified Integral
\( \int f(u)\,du \)
 Integrate
\( F(u) + C \)
Back-substitute: \( u = g(x) \)
\( F(g(x)) + C \)

Indefinite vs Definite Substitution

Indefinite Definite
Form \( \int f(g(x))\,g'(x)\,dx \) \( \int_a^b f(g(x))\,g'(x)\,dx \)
Limits No limits: back-substitute at end Convert limits: \( a \to g(a),\; b \to g(b) \)
Back-sub? Yes: replace \( u \to g(x) \) No: evaluate directly in \( u \)
Final answer \( F(g(x)) + C \) A number

How to Apply \( u \)-Substitution

1
Choose \( u \): Identify the inner function in the composition. Look for the part whose derivative also appears in the integrand.
2
Compute \( du \): Differentiate: \( du = g'(x)\,dx \). Solve for \( dx \) if necessary.
3
Rewrite: Replace all \( x \)-expressions with \( u \) and \( du \). No \( x \) should remain.
4
Integrate in \( u \): Evaluate \( \int f(u)\,du \) using standard formulas.
5
Back-substitute (indefinite only): Replace \( u \) with \( g(x) \) and add \( +C \).

Example 1: Basic Substitution

1
Let \( u = x^2 \)
2
Then \( du = 2x\,dx \)
3
Integral becomes: \( \int \cos(u)\,du \)
4
Integrate: \( \sin(u) + C \)
5
Back-substitute: \( \sin(x^2) + C \)

Example 2: Adjusting Constants

1
Let \( u = x^3 \), so \( du = 3x^2\,dx \)
2
The integrand has \( x^2\,dx \) but not \( 3x^2\,dx \). Solve: \( x^2\,dx = \frac{1}{3}\,du \)
3
Rewrite: \( \displaystyle\frac{1}{3}\int e^u\,du \)
4
Result: \( \dfrac{1}{3}e^{x^3} + C \)

Example 3: Definite Integral

1
Let \( u = x^2 + 1 \), so \( du = 2x\,dx \), i.e. \( x\,dx = \frac{1}{2}\,du \)
2
Change limits: when \( x=0 \), \( u=1 \); when \( x=2 \), \( u=5 \)
3
Rewrite: \( \displaystyle\frac{1}{2}\int_1^5 u^{1/2}\,du = \frac{1}{2}\cdot\frac{2}{3}u^{3/2}\Big|_1^5 \)
4
Result: \( \dfrac{1}{3}\bigl(5\sqrt{5} - 1\bigr) \)

Choosing \( u \) Wisely

Inside Argument
Pick the expression inside a trig, exponential, logarithmic, or radical: \( u = x^2+1 \) in \( \sin(x^2+1) \).
Derivative Present
Confirm that \( g'(x) \) (or a constant multiple) appears as a factor in the integrand.
Simplify Result
After substitution, the integral in \( u \) should match a known antiderivative formula.

Common Substitution Patterns

Integral Pattern \( u \) Choice Result
\( \int f(ax+b)\,dx \) \( u = ax+b \) \( \frac{1}{a}F(u)+C \)
\( \int x^n f(x^{n+1})\,dx \) \( u = x^{n+1} \) \( \frac{1}{n+1}\int f(u)\,du \)
\( \int \frac{f'(x)}{f(x)}\,dx \) \( u = f(x) \) \( \ln|u|+C \)
\( \int e^{g(x)}g'(x)\,dx \) \( u = g(x) \) \( e^u+C \)

Example 4: The \( \ln \) Pattern

1
Let \( u = x^2 + 3 \), so \( du = 2x\,dx \)
2
Integral becomes: \( \displaystyle\int \frac{1}{u}\,du \)
3
Result: \( \ln|x^2+3| + C = \ln(x^2+3) + C \)

Common Mistakes

✕ Forgetting \( du \)
Substituting \( u \) but leaving \( dx \) instead of converting to \( du \). Every \( dx \) must be replaced.
✕ Wrong Constant Factor
If \( du = 3x^2\,dx \), the leftover factor is \( \frac{1}{3} \), not \( 3 \). Divide, don't multiply.
✕ Keeping Old Limits
Using the original \( x \)-limits with a \( u \)-integral. Convert limits or back-substitute first.

Question 1 of 5

Evaluate \( \displaystyle\int 3x^2 \sin(x^3)\,dx \).

Question 2 of 5

To evaluate \( \displaystyle\int \frac{e^{\sqrt{x}}}{\sqrt{x}}\,dx \), which substitution should you use?

Question 3 of 5

Evaluate \( \displaystyle\int_0^1 \frac{x}{(x^2+1)^2}\,dx \).

Question 4 of 5

What is \( \displaystyle\int \tan(x)\,dx \) using substitution?

Question 5 of 5

A student writes \( \displaystyle\int_0^{\pi} \sin(2x)\,dx \) and substitutes \( u = 2x \), but evaluates with limits 0 to \( \pi \). What error did they make?