Change of Variables

The change of variables formula states that if u = g(x), then du = g(xdx and hence

$\displaystyle \int$f (g(x)) g(x) dx = $\displaystyle \int$f (u) du

To perform a change of variables on $\int$x sin x2dx

1.
Enter the integral $\int$x sin x2 dx.

2.
From the Calculus submenu, choose Change Variable.

3.
Enter the substitution u = x2 in the dialog box.

4.
Choose OK.

$\blacktriangleright$ Calculus + Change Variable

$\dint$x sin x2 dx = $\dint$${\frac{{1}}{{2}}}$sin u  du

In this example, u = x2. Note that u = g(x) = x2, f (u) = sin u, and du = 2x dx.

$\blacktriangleright$ Calculus + Change Variable

$\dint$x2$\sqrt{{x^{3}+1}}$dx = $\dint$${\frac{{2}}{{3}%
}}$u2 du = ${\frac{{2}}{{9}}}$u3 = ${\frac{{2}}{{9}}}$$\left(\vphantom{ \sqrt{%
x^{3}+1}}\right.$$\sqrt{{%
x^{3}+1}}$$\left.\vphantom{ \sqrt{%
x^{3}+1}}\right)^{{3}}_{}$

In this example, u2 = x3 + 1.