Are there any efficient ways to calculate this by hand? The integral appeared on a University engineering entrance exam to which I don't have the solutions. Putting it into an online integral calculator (https://www.integral-calculator.com/) gives about 60 lines of working leading to the answer $\frac{19\pi}{12}$. The technique used by the online calulator seems too long and difficult for what was just a part of a question. Any help would be much appreciated.
$$\int_{-6}^6{\frac{19+20\sin^7x}{x^2+36}}\,\mathrm dx$$
Best Answer
The indefinite integral is tremendous, but here you have a symmetrical range so that you should look at the parity of the integrand. For this reason, $$\int_{-6}^6{\frac{19+20\sin^7x}{x^2+36}}\,\mathrm dx=2\int_0^6{\frac{19}{x^2+36}}\,\mathrm dx=\frac{19}3\,\left.\arctan\frac x6\right|_0^6=\frac{19\pi}{12}.$$