Question

I am thinking about this integral
(x^2)*(sqrt (x^2+16))🔍
Can I rationalize it?
***
One more question about Ostogradski formulas -does it exist his formula for integrals such is
(ax^3+bx^2+cx+d)(sqrt...)+λ/(sqrt..)...?

Answer 1

Did you try a trig sub?

Answer 2

first one works great using trig sub

Answer 3

Yes, I take x=4tgt but I can't finish .😕

Answer 4

\[\begin{align*}\int x^2\sqrt{x^2+16}\,\mathrm{d}x&=\int (4\tan t)^2\sqrt{(4\tan t)^2+4^2}\,\mathrm{d}t\\[1ex]
&=16\sqrt{16}\int\tan^2t\sqrt{\tan^2t+1}\,\mathrm{d}t\\[1ex]
&=64\int \tan^2t\sec t\,\mathrm{d}t&(*)
\end{align*}\](\((*)\) for valid \(t\))

Hint: If you can integrate \(\sec^3t\), the rest should be easy.

Answer 5

\[\int\limits x^2 \sqrt{x^2+16} dx=\int\limits (4 \tan(t))^2 \sqrt{(4 \tan(t))^2+4^2} 4 \sec^2(t) dt \\ \text{ since } dx \neq dt \text{ but } dx=4 \sec^2(t) dt \\ \]

Answer 6

Whoops, my mistake. The last line should then be
\[256\int\tan^2t\sec^3t\,\mathrm{d}t\]Thanks @freckles