Find an appropriate trigonometric substitution of the form $x=f(t)$ to simplify the integral $ \int x\sqrt{7x^2+42x+59}\,\mathrm dx $

Question

Find an appropriate trigonometric substitution of the form $x=f(t)$ to simplify the integral $$ \int x\sqrt{7x^2+42x+59}\,\mathrm dx .$$

There were never any examples quite like this in class, so I'm clueless as to how to figure out which trig function to use.

Answer 1

Try completing the square to get

\begin{equation*} \int x\sqrt{(\sqrt{7}x+3\sqrt{7})^2-4)}\,\mathrm dx \end{equation*}

& use the substitution $u=\sqrt{7}x+3\sqrt{7},~du=\sqrt{7}dx.$