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Substitution — First Type (Differential Matching)

The first-type substitution (differential matching) is one of the most common methods.

Basic Idea

Set part of the integrand as a new variable to simplify:

Let $u=\varphi(x)$ , then $\int f(\varphi(x))\,\varphi'(x)\,dx = \int f(u)\,du.$

Steps

Identify the composite part.
Let $u=\varphi(x)$ , compute $du=\varphi'(x)\,dx$ .
Rewrite the integral in $u$ .
Integrate, then substitute back.

Examples

Example 1 $\int x e^{x^2} dx$ → $u=x^2$ , $du=2x\,dx$ → $\tfrac12 e^{x^2}+C$ .
Example 2 $\int \dfrac{1}{x \ln x} dx$ → $u=\ln x$ , $du=\dfrac{1}{x}dx$ → $\ln|\ln x|+C$ .
Example 3 $\int \sin(2x+1) dx$ → $u=2x+1$ , $du=2dx$ → $-\tfrac12\cos(2x+1)+C$ .

Practice Problems

Exercise 1

Use the substitution method to compute $\int x e^{x^2} dx$ .

Reference Answer

first-type substitutiondifferential matching

Approach: The integrand has the composite form $f(g(x))g'(x)$ : here $g(x)=x^2$ and $g'(x)=2x$ , so the factor $x\,dx$ supplies half of the missing differential. Set $u=x^2$ .

Steps:

Substitute $u=x^2$ , so $du=2x\,dx$ and $x\,dx=\tfrac12 du$ .
Rewrite the integral: $\int x e^{x^2} \, dx = \tfrac12 \int e^u \, du = \tfrac12 e^u + C$ .
Back-substitute: $\tfrac12 e^u + C = \tfrac12 e^{x^2} + C$ .

Answer: $\tfrac12 e^{x^2} + C$ .

Summary

Symbols Used in This Article

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$u$	variable	yoo	The new variable after substitution
$\int$	mathematical symbol	integral	Indefinite/definite integral sign
$\varphi(x)$	function symbol	phi of x	The composite inner function whose derivative is matched
$\varphi'(x)$	mathematical symbol	phi prime of x	The derivative of the inner function, supplying the differential
$du$	mathematical symbol	differential of u	The differential that replaces $\varphi'(x)\,dx$
$dx$	mathematical symbol	differential of x	The differential element of the integration variable $x$
$C$	mathematical symbol	constant of integration	The arbitrary constant restored in every antiderivative
$x^2$	mathematical symbol	x squared	The inner function chosen in Example 1
$\ln x$	mathematical symbol	natural logarithm of x	The inner function chosen in Example 2
$\ln\lvert\ln x\rvert$	mathematical symbol	natural logarithm of the absolute value of ln x	The antiderivative obtained in Example 2
$\sin(2x+1)$	mathematical symbol	sine of 2x plus 1	The integrand of Example 3, whose inner function is affine

Substitution (Integration)

Substitution — Second Type (Trigonometric)

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