Calculus I · Unit 3A · lesson

Definite Integrals by Substitution

Concept

Learning objectives

Change bounds consistently or back-substitute before applying original bounds.

Definite Integrals by Substitution

Explanation

Change the variable and the bounds together

In a definite integral, substitution changes not only the integrand but also the coordinate used to describe the interval. If $u=g(x)$ , convert the original $x$ -bounds into $u$ -bounds immediately. Then the entire calculation can remain in $u$ , avoiding a needless back-substitution before evaluating endpoints.

Mixing an integrand in $u$ with bounds in $x$ is a category error, not a minor notation blemish. The bounds must describe values of the current integration variable. After evaluation, interpret the answer in the original problem's units and context; the temporary variable is a computational device, while the definite integral still represents the same accumulated quantity.

For definite integrals, two valid methods exist:

• transform the integrand and change the bounds into $u$ -values; or • find an antiderivative in $u$ , return to $x$ , and use the original bounds.

Do not transform the integrand into $u$ while leaving $x$ -bounds attached.

Worked example

Change the bounds

\int_0^1 2x(x^2+1)^3\,dx.

Let $u=x^2+1$ , $du=2x\,dx$ . When $x=0$ , $u=1$ ; when $x=1$ , $u=2$ . Therefore

\int_1^2u^3\,du=\left[\frac{u^4}4\right]_1^2=\frac{15}{4}.

Interactive checku3a-def-sub-01

Evaluate $\int_0^1 x e^{x^2}\,dx$ .

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Show hint

Let $u=x^2$ , so $du=2x\,dx$ , and change the bounds to 0 and 1.

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After the explanation

Use the section idea

Reading lens

Choose an integration method from the integrand's structure, then verify the result by differentiation.

Mental model

Substitution reverses a chain rule, parts reverses a product rule, and algebraic or trigonometric rewrites expose a recognizable derivative pattern.

Decision

Simplify first; look for an inner derivative; then consider parts, identities, trigonometric substitution, or partial fractions in a deliberate order.

Common trap

Choosing a method by surface appearance, transforming only part of the differential, or accepting a more complicated integral than the one you started with.

Check yourself

Can you name the derivative rule being reversed and differentiate the final answer back to the integrand?

Source & rights

Original instruction with traceable references.

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