Question

Solve the indefinite integral: x^3(1+x)^(3/2)
Show steps.

Answer 1

Use integration by parts 3 times.
\[\int{x^3(1+x)^{3/2}}dx\]
Let u=x^3, v' = (1+x)^(3/2)
u' = 3x^2, v = (1+x)^(5/2) / (5/2)
\[\int{x^3(1+x)^{3/2}}dx=x^3\frac{(1+x)^{5/2}}{5/2}-\frac{3}{5/2}\int{x^2(1+x)^{5/2}}dx\]
Repeat this process twice more to get to your solution.

Answer 2

Thanks!

Answer 3

\[\int\limits x^3 (1+x)^{3/2}dx=\frac{2 (x+1)^{5/2} \left(105 x^3-70 x^2+40 x-16\right)}{1155} \]