MHB What is the solution to this week's challenging integral problem?

  • Thread starter Thread starter anemone
  • Start date Start date
  • Tags Tags
    2017
AI Thread Summary
The integral problem presented is to compute the definite integral from 0 to π of the function (2sin x + 3cos x - 3) divided by (13cos x - 5). Members are encouraged to refer to the guidelines for submitting solutions. Two users, Opalg and greg1313, successfully provided correct solutions to the problem. The discussion highlights the importance of collaboration in solving challenging mathematical problems. Engaging with such integrals can enhance problem-solving skills in calculus.
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
Here is this week's POTW:

-----

Compute $$\int_{0}^{\pi} \dfrac{2\sin x+3\cos x-3}{13\cos x-5} \,dx$$.

-----

Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
Physics news on Phys.org
Congratulations to the following members for their correct solution::)

1. Opalg
2. greg1313

Solution from Opalg:
At first sight, this looks like an improper integral, because the denominator vanishes when $\cos x = \frac5{13}$. But if $\cos x = \frac5{13}$ then (from the 5-12-13 pythagorean triple) $\sin x = \frac{12}{13},$ which means that $2\sin x + 3\cos x -3 = 0.$ So the numerator also vanishes at that point, and in fact there is no singularity.

To evaluate the integral, use the magical substitution $t = \tan\frac x2$. Then $\sin x = \frac{2t}{1+t^2}$, $\cos x = \frac{1-t^2}{1+t^2}$ and $dx = \frac2{1+t^2}\,dt$. So $$\frac{2\sin x + 3\cos x -3}{13\cos x - 5} = \frac{4t + 3(1-t^2) - 3(1+t^2)}{13(1-t^2) - 5(1+t^2)} = \frac{4t - 6t^2}{8-18t^2} = \frac{2t(2-3t)}{2(2-3t)(2+3t)} = \frac{t}{2+3t},$$ and the integral becomes $$\int_0^\pi \frac{2\sin x + 3\cos x -3}{13\cos x - 5}\,dx = \int_0^\infty\frac t{2+3t}\frac2{1+t^2}\,dt.$$ Next, use partial fractions to get $$\frac{2t}{(2+3t)(1+t^2)} = -\frac{12}{13}\frac1{2+3t} + \frac4{13}\frac t{1+t^2} + \frac6{13}\frac 1{1+t^2}.$$ That function has an indefinite integral $$f(t) = -\frac4{13}\ln(2+3t) + \frac2{13}\ln(1+t^2) + \frac6{13}\arctan t = \frac2{13}\ln\Bigl(\frac{1+t^2}{(2+3t)^2}\Bigr) + \frac6{13}\arctan t.$$

At the lower end of the interval of integration, $$f(0) = -\frac4{13}\ln2$$. At the upper end, $$\lim_{t\to\infty}\frac{1+t^2}{(2+3t)^2} = \frac19,$$ and so $$\lim_{t\to\infty}f(t) = \frac2{13}\ln\Bigl(\frac19\Bigr) + \frac6{13}\frac\pi2 = -\frac4{13}\ln3 +\frac{3\pi}{13}.$$

In conclusion, $$\int_0^\pi \frac{2\sin x + 3\cos x -3}{13\cos x - 5}\,dx = -\frac4{13}\ln3 +\frac{3\pi}{13} + \frac4{13}\ln2 = \frac{3\pi - 4\ln(3/2)}{13}.$$
 
Back
Top