u-Substitution Formula

U-substitution is an integration technique where you substitute u = g(x) and du = g'(x)\,dx to transform a complicated integral into a simpler one.

The Formula

∫f(g(x)) g′(x) dx=∫f(u) duwhere u=g(x)\int f(g(x))\,g'(x)\,dx = \int f(u)\,du \quad \text{where } u = g(x)
For definite integrals: change the bounds too! If u=g(x)u = g(x), then when x=ax = a, u=g(a)u = g(a).

When to use: When you see a composite function inside an integral along with its inner derivative lurking nearby, substitution collapses the composition into a single variable. It's like un-nesting a function: replace the inner part with uu, and the integral becomes simpler.

Quick Example

∫2xcos⁡(x2) dx\int 2x\cos(x^2)\,dx Let u=x2u = x^2, du=2x dxdu = 2x\,dx.
=∫cos⁡u du=sin⁡u+C=sin⁡(x2)+C= \int \cos u\,du = \sin u + C = \sin(x^2) + C

Notation

Let u=g(x)u = g(x), then du=g′(x) dxdu = g'(x)\,dx. After integrating in terms of uu, substitute back to express the result in terms of xx.

What This Formula Means

An integration technique where you substitute u=g(x)u = g(x) and du=g′(x) dxdu = g'(x)\,dx to transform a complicated integral into a simpler one. It is the reverse of the chain rule for differentiation.

When you see a composite function inside an integral along with its inner derivative lurking nearby, substitution collapses the composition into a single variable. It's like un-nesting a function: replace the inner part with uu, and the integral becomes simpler.

Formal View

If gg is differentiable on [a,b][a, b] and ff is continuous on the range of gg, then ∫abf(g(x))⋅g′(x) dx=∫g(a)g(b)f(u) du\int_a^b f(g(x)) \cdot g'(x)\,dx = \int_{g(a)}^{g(b)} f(u)\,du where u=g(x)u = g(x).

Worked Examples

Example 1

easy
Find ∫3x2(x3+1)4 dx\displaystyle\int 3x^2(x^3+1)^4\,dx.

Answer

(x3+1)55+C\frac{(x^3+1)^5}{5} + C

First step

1
Let u=x3+1u = x^3+1, so du=3x2 dxdu = 3x^2\,dx.

Full solution

  1. 2
    Integral becomes ∫u4 du=u55+C\int u^4\,du = \frac{u^5}{5} + C.
  2. 3
    Substitute back: (x3+1)55+C\frac{(x^3+1)^5}{5} + C.
3x2 dx3x^2\,dx is exactly dudu, so the substitution is perfect. After substitution the integral reduces to a simple power rule.

Example 2

medium
Evaluate ∫01xex2 dx\displaystyle\int_0^1 xe^{x^2}\,dx.

Example 3

medium
Evaluate ∫x2x3+1 dx\displaystyle\int x^2 \sqrt{x^3+1}\,dx.

Common Mistakes

  • Forgetting to convert dxdx via du=g′(x) dxdu=g'(x)\,dx — you must replace dxdx too, not just the inner expression.
  • Leaving the answer in terms of uu for an indefinite integral — substitute back to xx at the end.
  • Not changing the bounds for a definite integral — either switch limits to uu-values or convert back before evaluating.

Why This Formula Matters

u-Substitution is the most-used integration technique and the inverse of the chain rule, which is why most antiderivatives of composites depend on it. The skill it builds — spotting that g′(x) dxg'(x)\,dx is hiding next to g(x)g(x) — is exactly the chain-rule structure read backward, and forgetting to change the bounds (or the dxdx) is the classic slip. Recognizing it by "Is there an inner function g(x)g(x) inside, with its derivative g′(x)g'(x) also present as a factor?" — rather than by familiar numbers — is what lets a student tell it apart from chain rule and integration by parts and direct antiderivative in a mixed problem set.

Frequently Asked Questions

What is the u-Substitution formula?

An integration technique where you substitute u=g(x)u = g(x) and du=g′(x) dxdu = g'(x)\,dx to transform a complicated integral into a simpler one. It is the reverse of the chain rule for differentiation.

How do you use the u-Substitution formula?

When you see a composite function inside an integral along with its inner derivative lurking nearby, substitution collapses the composition into a single variable. It's like un-nesting a function: replace the inner part with uu, and the integral becomes simpler.

What do the symbols mean in the u-Substitution formula?

Let u=g(x)u = g(x), then du=g′(x) dxdu = g'(x)\,dx. After integrating in terms of uu, substitute back to express the result in terms of xx.

Why is the u-Substitution formula important in Math?

u-Substitution is the most-used integration technique and the inverse of the chain rule, which is why most antiderivatives of composites depend on it. The skill it builds — spotting that g′(x) dxg'(x)\,dx is hiding next to g(x)g(x) — is exactly the chain-rule structure read backward, and forgetting to change the bounds (or the dxdx) is the classic slip. Recognizing it by "Is there an inner function g(x)g(x) inside, with its derivative g′(x)g'(x) also present as a factor?" — rather than by familiar numbers — is what lets a student tell it apart from chain rule and integration by parts and direct antiderivative in a mixed problem set.

What do students get wrong about u-Substitution?

The procedure for u-substitution is the easy part; the trap is forgetting to convert dxdx via du=g′(x) dxdu=g'(x)\,dx. Asking "Is there an inner function g(x)g(x) inside, with its derivative g′(x)g'(x) also present as a factor?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the u-Substitution formula?

Before studying the u-Substitution formula, you should understand: integral, chain rule.

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How to Integrate Rational Functions: Long Division and Partial Fractions →
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