# Antiderivatives by substitution of variables

Antidifferentiation of a Composite Function
$\int\ F'(g(x))g'(x)\,dx=F(g(x))+C=F(u)+C.$

Change of Variables
With a formal change of variables, you completely rewrite the integral in terms of u and du (or any other convenient variable). The change of variable technique uses the Leibniz notation for the differential. That is, it u=g(x), then du=g'(x)dx, and the integral in Theorem 4.12(shown above) takes the form:
$\int\ f(g(x))g'(x)\,dx=\int\ f(u)\,du=F(u)+C.$
Example
 $\int\ \sqrt{2x-1}\,dx.$ Original Function. $\int\ \sqrt{u}\,(\frac{du}{2}).$ Write Integral in terms of u. $\frac{1}{2}\ \int\ u^{\frac{1}{2}} ,du.$ Constant Multiple Rule. $\frac{1}{2}\ (\frac{u^{\frac{3}{2}}}{\frac{3}{2}}}) + C.$ Write the antiderivative in terms of u. $\frac{1}{3}\ u^{\frac{3}{2}} + C.$ Simplify. $\frac{1}{3}\ (2x-1)^{\frac{3}{2}} +C.$ Replace u terms with x terms.

Guidelines for Making a Change of Variables
• 1. Choose a substitution u=g(x). Usually, it is best to choose the inner part of a composite function, such as a quantity raised to a power.
• 2. Compute du=g'(x) dx.
• 3. Rewrite the integral in terms of the variable u.
• 4. Find the resulting integral in terms of u.
• 5. Replace u by g(x) to obtain an antiderivative in terms of x.