Solve Integration Problem: ∫x^3/(x^2+1)^(3/2) dx

[tmwong]

can anyone solve this equation?

∫x^3/(x^2+1)^(3/2) dx

 

[NeutronStar]

$$\int\frac{x^3}{\left(x^2+1\right)^{\frac{3}{2}}}dx=\frac{x^2+2}{\sqrt{x^2+1}}+C$$

 

[Galileo]

Hmm, maybe integration by parts would work?
$$\frac{x^3}{\left(x^2+1\right)^{\frac{3}{2}}}=x^2\frac{x}{\left(x^2+1\right)^{\frac{3}{2}}}$$
The primitive of the factor on the right is
$$\frac{-1}{\sqrt{x^2+1}}$$
looks like it may lead somewhere.
Appears tedious though

 

[sally]

 

[dextercioby]

$$\int\frac{x^3}{\left(x^2+1\right)^{\frac{3}{2}}}dx=<br /> \frac{x^2+2}{\sqrt{x^2+1}}+C$$

Proof
$$\int\frac{x^3}{\left(x^2+1\right)^{\frac{3}{2}}}dx=<br /> \int\frac{x^2}{2}\frac{2x}{(\sqrt{x^2+1})^3}dx=<br /> -\frac{x^2}{\sqrt{x^2+1}}+\int\frac{2x}{\sqrt{x^2+1}}dx=<br /> -\frac{x^2}{\sqrt{x^2+1}}+2\sqrt{x^2+1}+C=<br /> \frac{x^2+2}{\sqrt{x^2+1}}+C$$