Chapter 13 - Section 13.2 - Substitution - Exercises - Page 971: 37

$\int 2x\sqrt {x+1}dx=\frac{4}{5}(x+1)^{\frac{5}{2}}-\frac{4}{3}(x+1)^{\frac{3}{2}}+C$

Substitution: $u=x+1$ $x=u-1$ $dx=du$ $\int 2x\sqrt {x+1}dx=\int2(u-1)u^{\frac{1}{2}}du=\int2(u^{\frac{3}{2}}-u^{\frac{1}{2}})du=2(\frac{u^{\frac{5}{2}}}{\frac{5}{2}}-\frac{u^{\frac{3}{2}}}{\frac{3}{2}})+C=\frac{4}{5}(x+1)^{\frac{5}{2}}-\frac{4}{3}(x+1)^{\frac{3}{2}}+C$