Integration By Substitution

Integration by Substitution

Integration by substitution is the reverse of the chain rule. It transforms integrals of the form $\int f(g(x))\,g'(x)\,dx$ into simpler ones by letting $u = g(x)$, so $du = g'(x)\,dx$, yielding $\int f(u)\,du.$

Choosing Your Substitution

The key is spotting the inner function — typically something nested inside a power, root, trigonometric, exponential, or logarithmic function. Check whether its derivative (or a constant multiple of it) appears elsewhere in the integrand. If it does, set $u$ equal to that inner function, compute the differential $du$, and rewrite the entire integral in terms of $u$ only. No $x$ variables should remain.

Indefinite Integrals

After integrating with respect to $u$, back-substitute $u = g(x)$ to express your answer in the original variable.

Example: For $\int 2x\cos(x^2)\,dx$, let $u = x^2$, so $du = 2x\,dx$: $\int \cos(u)\,du = \sin(u) + C = \sin(x^2) + C.$

Definite Integrals

For $\int_a^b f(g(x))g'(x)\,dx$, transform the bounds directly: when $x=a$, $u=g(a)$; when $x=b$, $u=g(b)$. Then evaluate entirely in $u$, avoiding the need to back-substitute.

Example: For $\int_0^2 x e^{x^2}\,dx$, let $u = x^2$, so $x\,dx = \frac{1}{2}du$. Bounds: $x=0 \Rightarrow u=0$; $x=2 \Rightarrow u=4$: $\frac{1}{2}\int_0^4 e^u\,du = \frac{1}{2}(e^4 - 1).$

Practical Note

Substitution simplifies integrals arising in applications like the Net Change Theorem. When velocity involves composite functions, substitution finds the antiderivative needed to compute displacement or distance.