U-Substitution Master Examples Cheat Sheet

A printable reference covering $u$-substitution, differential matching, definite integrals, and common substitution patterns for grades 11-12.

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This cheat sheet covers how to use $u$ -substitution to evaluate integrals by reversing the chain rule. Students need it because many integrals look complicated until the inside function and its derivative are matched. The examples help students choose $u$ , rewrite $dx$ , change bounds when needed, and avoid common algebra errors. It is designed as a formula-forward reference for grades 11-12 calculus practice. The core idea is to set $u = g(x)$ when an integral contains a composite function such as $f(g(x))g'(x)$ . Then compute $du = g'(x)\,dx$ and rewrite the whole integral in terms of $u$ . For indefinite integrals, integrate in $u$ and substitute back to $x$ . For definite integrals, either change the bounds to $u$ -values or substitute back before applying the original $x$ -bounds.

Key Facts

The basic $u$ -substitution pattern is $\int f(g(x))g'(x)\,dx = \int f(u)\,du$ when $u = g(x)$ and $du = g'(x)\,dx$ .
For an indefinite integral, after finding $\int f(u)\,du$ , substitute $u = g(x)$ back into the answer and add $+C$ .
For a definite integral, if $u = g(x)$ , then $\int_a^b f(g(x))g'(x)\,dx = \int_{g(a)}^{g(b)} f(u)\,du$ .
A constant multiplier can be adjusted because $\int f(g(x))g'(x)\,dx$ may require multiplying and dividing by the same constant.
For powers, the pattern $\int (g(x))^n g'(x)\,dx = \frac{(g(x))^{n+1}}{n+1}+C$ works when $n \ne -1$ .
For logarithmic forms, $\int \frac{g'(x)}{g(x)}\,dx = \ln|g(x)|+C$ .
For exponential forms, $\int e^{g(x)}g'(x)\,dx = e^{g(x)}+C$ .
For trigonometric forms, examples include $\int \cos(g(x))g'(x)\,dx = \sin(g(x))+C$ and $\int \sin(g(x))g'(x)\,dx = -\cos(g(x))+C$ .

Vocabulary

$u$ -substitution: A method for rewriting an integral using $u = g(x)$ so the integral becomes simpler to evaluate.
Composite function: A function inside another function, such as $\sin(3x^2)$ where $3x^2$ is the inside function.
Differential: The expression $du = g'(x)\,dx$ that tells how a small change in $u$ relates to a small change in $x$ .
Antiderivative: A function $F(x)$ whose derivative is the integrand, so $F'(x) = f(x)$ .
Definite integral bounds: The lower and upper limits of integration, which must be changed from $x$ -values to $u$ -values when using $u$ -substitution directly.
Constant of integration: The $+C$ added to an indefinite integral because antiderivatives differ by a constant.

Common Mistakes to Avoid

Choosing only the outside function for $u$ is wrong because $u$ should usually be the inside expression $g(x)$ whose derivative also appears in the integrand.
Forgetting to rewrite $dx$ using $du$ is wrong because the integral must be completely converted into the new variable before integrating.
Dropping a constant factor is wrong because if $du = 6x\,dx$ but the integral has $2x\,dx$ , the substitution needs the factor $\frac{1}{3}$ .
Using the original bounds after changing to $u$ is wrong because bounds like $x = a$ and $x = b$ must become $u = g(a)$ and $u = g(b)$ .
Forgetting to substitute back in an indefinite integral is wrong because the final antiderivative must be written in terms of the original variable unless the problem says otherwise.

Practice Questions

1 Evaluate $\int 6x(3x^2+5)^4\,dx$ using $u$ -substitution.
2 Evaluate $\int \frac{2x}{x^2+9}\,dx$ using $u$ -substitution.
3 Evaluate $\int_0^1 4x e^{2x^2}\,dx$ by changing the bounds to $u$ -values.
4 Explain why $u = x^2+1$ is a better substitution than $u = \sqrt{x^2+1}$ for an integral containing $\frac{x}{\sqrt{x^2+1}}\,dx$ .

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