U-substitution is the integral version of "undoing" the chain rule. It's usually the first integration technique students learn in Calc II, and it works well once you can recognize the pattern — which is the part most students struggle with at first.
When to use it
Look for an integral where one part of the expression is the derivative of another part (or a constant multiple of it). A common giveaway: a function "inside" another function, multiplied by something that looks like the derivative of the inside piece.
The step-by-step method
- Choose u — usually the "inside" function (what's inside parentheses, under a root, or in an exponent).
- Find du by differentiating u with respect to x, then write du = ... dx.
- Rewrite the entire integral in terms of u, replacing every x-term, including dx.
- Integrate with respect to u using standard rules.
- Substitute back to express the answer in terms of x (unless you already changed the bounds — see below).
Worked example
Evaluate ∫ 2x·cos(x²) dx
Let u = x², so du = 2x dx — which conveniently matches the 2x already in the integral.
The integral becomes ∫ cos(u) du = sin(u) + C
Substituting back: sin(x²) + C
Common mistakes
Forgetting to substitute dx. Every part of the original integral needs to be rewritten in terms of u, including dx — not just the obvious "inside function" piece.
Not changing the bounds on a definite integral. If you're evaluating from a to b, either convert the bounds to u-values right when you substitute, or make sure to substitute back to x before plugging in the original bounds. Mixing the two approaches is a very common error.
Stuck on a calculus problem right now?
Tell us what you're working on and get matched with a tutor who explains the reasoning, not just the answer — free quote by email or WhatsApp.
Get a free quote