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u-Substitution — Definition, Formula & Examples

u-Substitution
Substitution Method
Integration by Substitution

An integration method that essentially involves using the chain rule in reverse.

Example: For ∫x5+x2 dx\displaystyle \int x\sqrt{5+x^2}\,dx let u=5+x2.u=5+x^2\text{.}
That means du=2x dx\displaystyle du = 2x\,dx
12du=x dx\displaystyle \frac{1}{2}du = x\,dx
It follows that∫x5+x2 dx=∫u⋅12du=12∫u12 du=12⋅23u32+C=13(5+x2)32+C\begin{aligned} \int x\sqrt{5+x^2}\,dx &= \int\sqrt{u}\cdot\frac{1}{2}du \\ &= \frac{1}{2}\int u^{\frac{1}{2}}\,du \\ &= \frac{1}{2}\cdot\frac{2}{3}u^{\frac{3}{2}}+C \\ &= \frac{1}{3}\left(5+x^2\right)^{\frac{3}{2}}+C \end{aligned}

Key Formula

∫f(g(x)) g′(x) dx=∫f(u) duwhere u=g(x)\int f\bigl(g(x)\bigr)\, g'(x)\, dx = \int f(u)\, du \quad \text{where } u = g(x)
Where:
  • uu = The substitution variable, set equal to an inner function g(x)
  • g(x)g(x) = An inner function of x chosen to simplify the integral
  • g′(x)g'(x) = The derivative of g(x), which must appear (up to a constant) in the integrand
  • f(u)f(u) = The outer function, now written in terms of u

Worked Example

Problem: Evaluate the integral ∫2xcos⁡(x2) dx\int 2x \cos(x^2)\, dx.
Step 1: Choose u to be the inner function. Let u=x2u = x^2.
u=x2u = x^2
Step 2: Differentiate u with respect to x and solve for dx.
du=2x dx⟹dx=du2xdu = 2x\, dx \quad \Longrightarrow \quad dx = \frac{du}{2x}
Step 3: Substitute into the integral. The 2x2x in the integrand cancels with the 2x2x in the denominator of dxdx.
∫2xcos⁡(x2) dx=∫cos⁡(u) du\int 2x \cos(x^2)\, dx = \int \cos(u)\, du
Step 4: Integrate with respect to u.
∫cos⁡(u) du=sin⁡(u)+C\int \cos(u)\, du = \sin(u) + C
Step 5: Substitute back, replacing u with x2x^2.
sin⁡(x2)+C\sin(x^2) + C
Answer: sin⁡(x2)+C\sin(x^2) + C

Why It Matters

u-Substitution is the most frequently used integration technique in calculus. Many integrals that look difficult become straightforward once you identify the right substitution. Mastering it is essential before moving on to more advanced methods like integration by parts or partial fractions.

Common Mistakes

Mistake: Forgetting to convert every part of the integrand—including dxdx—into terms of uu and dudu.
Correction: After choosing uu, compute dudu and replace dxdx accordingly. No xx terms should remain in the integral once the substitution is complete.

Related Terms