Integration by Substitution Type 1¶

Proposition¶

Let $g\colon[\alpha,\beta]\to\mathbb{R}$ be continuously differentiable, and let $f$ be continuous on an interval containing the range of $g\text{.}$ Then

Interpretation¶

Substitution of type 1 is the chain rule read backwards. When the integrand has the form $f(g(x))g'(x)$, the inner function $g(x)$ is renamed $t$, and the integral becomes $\int f(t)\,\mathrm{d}t$ in the new variable. It works best when the derivative factor $g'(x)$ already appears in the integrand.

The colours in the formula show the matching. The red part $f(g(x))$ becomes $f(t)$, and the blue part $g'(x)\,\mathrm{d}x$ becomes $\mathrm{d}t$. As $x$ runs from $\alpha$ to $\beta$, $t=g(x)$ runs from $g(\alpha)$ to $g(\beta)$, so the limits change too and there is no need to return to $x$ at the end.

The equality concerns Riemann integrals under the stated continuity assumptions. The differential relation $\mathrm{d}t=g'(x)\,\mathrm{d}x$ records the derivative factor, and $t=g(x)$ gives the transformed endpoints. The hypotheses make both sides ordinary integrals of continuous functions, and $f$ has an antiderivative $F$ with $(F\circ g)'=f(g(x))g'(x)$$f(g(x))g^{\prime }(x)$. The function $g$ may go back and forth: for $g(x)=\sin x$ on $[0,\pi]$, $g(0)=g(\pi)=0$$0$, so $\int_0^\pi f(\sin x)\cos x\,\mathrm{d}x=0$ for every continuous $f$.

A common slip is to substitute when the factor $g'(x)$ is missing. The substitution $t=x^2$ turns $\int 2x\cos(x^2)\,\mathrm{d}x$ into $\int\cos t\,\mathrm{d}t$, but it does not help with $\int\cos(x^2)\,\mathrm{d}x$. Another is to change the integrand but keep the old limits $\alpha,\beta$.

Key Results¶

• The function $g$ need not be invertible; the formula follows from the chain rule and the fundamental theorem of calculus.
• The limits of integration must be transformed together with the integrand, from $\alpha,\beta$ to $g(\alpha),g(\beta)\text{.}$
• If $F'=f\text{,}$ then the corresponding indefinite identity is $\displaystyle\int f(g(x))g'(x)\,\mathrm{d}x=F(g(x))+C\text{.}$

Examples¶

• $\displaystyle \int 2x\cos(x^2)\,\mathrm{d}x$ can be simplified by proposing $t=x^2\text{.}$
• $\displaystyle\int_1^2 \frac{3x^2+2}{x^3+2x+3}\,\mathrm{d}x$ can be evaluated using $t=x^3+2x+3\text{.}$

Prerequisites¶

• Riemann Integral
• Differentiability Class
• Continuity
• Chain Rule
• Differential

Related Concepts¶

• Integration by Substitution Type 2
• Indefinite Integral
• Integration by Parts

Sources¶

• Math I: Calculus · Lecture 12

Aliases: u-substitution Keywords: change of variables, substitution rule Symbols: alpha, beta, element of, integral, pi, real numbers

Last updated: 27 October 2026 11:52