Calculus I · Unit 3A · lesson

Definite Integrals by Substitution

Learn Definite Integrals by Substitution through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.

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Section overview

Computing integrals

What this section is building

Learn Definite Integrals by Substitution through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.

Notice

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

Decide

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

Avoid

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

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

Your work stays on this device. No account or AI grader is used.

Show hint

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

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Source & rights

Original instruction with traceable references.

BetterGrades-original; no direct adaptation declared in the verified handoff.

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

Definite Integrals by Substitution

Explanation

Change the variable and the bounds together

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

Your work stays on this device. No account or AI grader is used.

Show hint

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

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Source & rights

Original instruction with traceable references.

BetterGrades-original; no direct adaptation declared in the verified handoff.

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

Definite Integrals by Substitution

Related resources

Original instruction with traceable references.

Definite Integrals by Substitution

Related resources

Original instruction with traceable references.