Integration by Substitution

Definition

Integration by substitution is a technique based on the chain rule:

$$\frac{d}{dx}f(g(x))=f^{\prime }(g(x))g^{\prime }(x).$$

It transforms an integral of a composite function into a simpler integral by introducing a substitution $u=g(x)$.

Intuition

Just as the chain rule tells us how to differentiate a composition, substitution is the chain rule run backwards. By renaming $g(x)$ as $u$, a complicated integrand is rewritten in a new variable where a standard form is visible. For definite integrals, the limits travel with the substitution — they convert to values of $u$, so we never need to undo the substitution at the end.

Formal Description

Given an integral of the form $\int f^{\prime }(g(x))g^{\prime }(x)dx$, let $u=g(x)$ with differential $du=g^{\prime }(x)dx$. Then

$$\int f^{\prime }(g(x))g^{\prime }(x)dx=\int f^{\prime }(u)du=f(u)+C=f(g(x))+C.$$

For definite integrals, the limits of integration must be transformed accordingly:

$${\int }_a^b{f}^{\prime }(g(x))g^{\prime }(x)dx={\int }_{g(a)}^{g(b)}{f}^{\prime }(u)du=f(g(b))-f(g(a)).$$

Example 1. $\int x{e}^{-x^2}dx$. Let $u=x^2$, $du=2xdx$:

$$\int x{e}^{-x^2}dx=\frac{1}{2}\int e^{-u}du=-\frac{1}{2}{e}^{-u}+C=-\frac{1}{2}{e}^{-x^2}+C.$$

Example 2. ${\int }_0^{\pi /2}{\sin }^3\theta \cos \theta d\theta$. Let $u=\sin \theta$, $du=\cos \theta d\theta$; limits change to $u=0$ and $u=1$:

$${\int }_0^{\pi /2}{\sin }^3\theta \cos \theta d\theta ={\int }_0^1{u}^{3}du=\frac{u^4}{4}{|}_0^1=\frac{1}{4}.$$

Applications

• Simplifying integrals of composite functions wherever $g^{\prime }(x)$ appears as a factor.
• Evaluating trigonometric, exponential, and logarithmic integrals by choosing $u$ to be the inner function.
• Converting definite integrals over one interval to equivalent integrals over a transformed interval.

Trade-offs

• The factor $g^{\prime }(x)$ must appear (possibly up to a constant multiple) in the integrand; substitution fails if it is absent.
• Choosing the right $u$ is not algorithmic — experience and pattern recognition are required.
• For definite integrals, forgetting to transform the limits is a common error; alternatively, back-substituting before evaluation avoids this but adds steps.
• Substitution handles one layer of composition at a time; nested compositions may require iterated substitutions.

Links

• Chain Rule
• Integrals
• Trigonometric Integrals