Chapter 6 - Inverse Functions - 6.4 Derivatives of Logarithmic Functions - 6.4 Exercises - Page 438: 82

$\int x2^{x^{2}}dx=\frac{1}{2}\frac{2^{{x^{2}}}}{ln2}+constant$

Evaluate the integral $\int x2^{x^{2}}dx$. Consider $x^{2}=t$ and $\frac{d}{dx}x^{2}=dt$ $2xdx=dt$ $xdx=\frac{dt}{2}$ Now, $\int x2^{x^{2}}dx=\int 2^{t}\frac{dt}{2}$ $=\frac{1}{2}\frac{2^{t}}{ln2}+constant$ Hence, $\int x2^{x^{2}}dx=\frac{1}{2}\frac{2^{{x^{2}}}}{ln2}+constant$