I have to calculate this integral:
\begin{align} \int x^{2}\sqrt{a^{2}+x^{2}}\,dx \qquad\text{with}
\quad a \in \mathbb{R} \end{align}
My attempt:
Using, trigonometric substitution
\begin{align}
\tan \theta &= \frac{x}{a}\\ \Longrightarrow \ x&=a \tan \theta\\ \Longrightarrow \ dx&=a \sec^{2}\theta\\ \Longrightarrow \ x^{2}&=a^{2}\tan^{2}\theta
\end{align}
Thus,
\begin{align}
\int x^{2}\sqrt{a^{2}+x^{2}}\,dx&=\int a^2 \tan^{2}\theta \sqrt{a^2+a^2\tan^{2}\theta}\ a\sec^{2}\theta\, d \theta\\&=a^{3}\int \tan^{2}\theta \sqrt{a^{2}(1+\tan^{2}\theta)}\sec^{2}\theta\, d\theta\\&=a^{3}\int \tan^{2}\theta \sqrt{a^{2}(\sec^{2}\theta)}\sec^{2}\theta \, d\theta\\&=a^{4}\int (1-\sec^{2}\theta)\sec^{3}\theta \, d\theta\\&=a^{4}\underbrace{\int \sec^{3}\theta \, d\theta}_{\text{solve by parts}}-a^{4}\underbrace{\int \sec^{5}\theta \, d\theta}_{\text{solve by parts}}
\end{align}
My doubt is: Is there any other way to solve it faster? Because by parts is a large process to solve each one. I really appreciate your help
Best Answer
$$x^2 \sqrt{a^2+x^2};\;\, x\to a \sinh u$$ $$dx=\cosh u\,du$$ $$\int x^2 \sqrt{a^2+x^2}\,dx=\int (a^2 \sinh ^2 u)(a \cosh u )\sqrt{a^2 \sinh ^2 u+a^2}\,du=$$ $$=a^4\int \sinh^2 u\cosh^2 u\,du=\frac{a^4}{4}\int\sinh^2 2u\,du=\frac{a^4}{8} \int (\cosh 4 u-1) \, du=$$ $$=\frac{1}{8} a^4 \left(\frac{1}{4} \sinh 4 u-u\right)+C=\frac{1}{8} \left(x \sqrt{a^2+x^2} \left(a^2+2 x^2\right)-a^4 \text{arcsinh}\left(\frac{x}{a}\right)\right)+C$$
Useful formulas
$\cosh^2 u -\sinh^2 u=1$
$\sinh 2u=2\sinh u\cosh u$
$\cosh 4u =\sinh ^2 2u +\cosh ^2 2u=2\sinh^2 2u+1\to \sinh^2 2u = \frac{1}{2}(\cosh4u - 1)$
$x=a\sinh u\to u=\text{arcsinh}\left(\frac{x}{a}\right)$
$\sinh 4u = 2\sinh 2u \cosh 2u = 4\sinh u\cosh u(\cosh^2 u+ \sinh^2 u)$