Evaluate an integral by substitution

Primary Tool

Math Workspace (CAS)

CAS is a good fit because the page can show the derivative of the inner expression, the rewritten integral in the new variable, and the final verification in one short transcript.

Problem

Evaluate $\int 2 x cos (x^{2}) d x$ .

The key structural cue is that the inner expression $x^{2}$ has derivative $2 x$ , and that derivative is already sitting in front of the cosine.

What the substitution is doing

The choice $u = x^{2}$ is not a guess pulled from nowhere. It is motivated by the structure of the integrand: one part is a composite function $cos (x^{2})$ and the other part is the derivative of the inside.

Once $u = x^{2}$ and $d u = 2 x d x$ , the original integral becomes $\int cos (u) d u$ , which is much easier to evaluate directly.

Step-by-step walkthrough

The goal is to replace a composite-looking integral in $x$ with a simpler integral in a new variable.

1Choose the substitution $u = x^{2}$ because the derivative of $x^{2}$ is $2 x$ .
2Differentiate to get $d u = 2 x d x$ .
3Rewrite the integral as $\int cos (u) d u$ .
4Integrate in the new variable to obtain $sin (u) + C$ .
5Substitute back $u = x^{2}$ to get the antiderivative $sin (x^{2}) + C$ .
6Differentiate $sin (x^{2})$ to check that it returns $2 x cos (x^{2})$ .

Common Pitfall

Substitution is not just renaming the variable. The replacement only works when you also transform the differential correctly, which is why $d u = 2 x d x$ is the decisive step.

Try a Variation

Try the same method on $\int 3 x^{2} e^{x^{3}} d x$ . Which inner expression should become $u$ , and what antiderivative do you get after substituting back?

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