U-Substitution

Reversing the Chain Rule

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U-substitution is a calculus technique for rewriting an integral so it becomes easier to evaluate. It is especially useful when an integrand contains a function and its derivative, or something close to that pattern. By introducing a new variable $u$ for the inner expression, you can turn a messy integral into a familiar basic form. This matters because many important integrals in physics, engineering, and mathematics depend on recognizing these hidden structures.

The method works by choosing $u = g(x)$ , then replacing $dx$ using $du = g'(x)\,dx$ . After substitution, the integral is written entirely in terms of $u$ , integrated with standard rules, and then converted back to $x$ . For definite integrals, the limits should also be changed from $x$ -values to $u$ -values so no back-substitution is needed. Success with u-substitution depends on spotting the inner function and checking whether its derivative is present as a factor or can be adjusted by a constant.

Key Facts

Choose the inner expression as the substitution: $u = g(x)$
Differentiate to relate the variables: $du = g'(x)\,dx$
Basic substitution rule: integral of $f(g(x))g'(x)\,dx = \int f(u)\,du$
Example: if $u = x^2 + 1$ , then $du = 2x\,dx$ , so $x\,dx = \frac{du}{2}$
For definite integrals, change limits using the substitution: if $x = a$ , then $u = g(a)$ ; if $x = b$ , then $u = g(b)$
After integrating in $u$ , substitute back to $x$ unless the integral is definite with updated limits

Vocabulary

Substitution: A method of replacing part of an expression with a new variable to simplify an $\int$ .
Inner function: The expression inside another function, such as $x^2 + 1$ inside $(x^2 + 1)^5$ .
Differential: A symbolic quantity like dx or du that shows the variable of integration and changes during substitution.
Antiderivative: A function whose derivative equals the original integrand.
Definite integral: An integral with upper and lower limits that represents accumulated change over an interval.

Common Mistakes to Avoid

Choosing $u$ without checking for $du$ , which is wrong because the derivative of the chosen expression must appear in the integral or be created with a constant factor.
Substituting only part of the integrand, which is wrong because after the change of variable the entire integral must be written in terms of $u$ and $du$ only.
Forgetting to change the limits in a definite integral, which is wrong because mixing $x$ -limits with a $u$ -integral leads to inconsistent work and wrong answers.
Not substituting back to $x$ in an indefinite integral, which is wrong because the final antiderivative should be expressed in the original variable unless new limits were used.

Practice Questions

1 Evaluate the integral of $2x(x^2 + 3)^4\,dx$ .
2 Evaluate the integral of $\frac{x}{x^2 + 1}\,dx$ .
3 Explain why $u = x^2$ is a good substitution for the integral of $x \cos(x^2)\,dx$ , and identify what part of the integrand becomes $du$ .