0

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.

$$\int_{-6}^6{\frac{19+20\sin^7x}{x^2+36}}\,\mathrm dx$$

Maximilian Janisch
  • 13,536
Rob-bxl
  • 3
  • Hint: Note that $$\frac{20\sin^7 x}{x^2+36}$$ is odd. Actually, this integral can be calculated very conveniently afterwards using the substitution $y=\frac x6$ – Maximilian Janisch Dec 10 '19 at 12:47
  • Here is a full solution that I don't want to post as an answer: $$I=19\int_{-6}^6\frac{1}{x^2+36},\mathrm dx = \frac{19}6\int_{-1}^1 \frac1{y^2+1},\mathrm dy = \frac{19}6[\arctan(y)]^1_{-1}=\frac{19}3 \cdot \frac\pi4$$ – Maximilian Janisch Dec 10 '19 at 12:55
  • A lot of problems are simplified by throwing away an integrand's odd part. Here is another question that does that. – J.G. Dec 10 '19 at 12:57
  • See https://math.stackexchange.com/questions/1073120/integral-int-12011-frac-sqrtx-sqrt2012-x-sqrtxdx OR https://math.stackexchange.com/questions/439851/evaluate-the-integral-int-frac-pi2-0-frac-sin3x-sin3x-cos3x?noredirect=1&lq=1 – lab bhattacharjee Dec 10 '19 at 13:28

1 Answers1

3

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, $$\int_{-6}^6{\frac{19+20\sin^7x}{x^2+36}}\,\mathrm dx=2\int_0^6{\frac{19}{x^2+36}}\,\mathrm dx=\frac{19}3\,\left.\arctan\frac x6\right|_0^6=\frac{19\pi}{12}.$$