Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Please help me to evaluate the following integral: $$\int\dfrac{\cos^{n-1}\dfrac{x+a}{2}}{\sin^{n+1}\dfrac{x-a}{2}}\;dx$$

share|cite|improve this question
up vote 1 down vote accepted

$$\int\frac{\cos^{n-1}\frac{x+a}{2}}{\sin^{n+1}\frac{x-a}{2}}dx=\int\frac{\cos^{n-1}\frac{x+a}{2}}{\sin^{n-1}\frac{x-a}{2}}\cdot\frac{dx}{\sin^2\frac{x-a}{2}}=\left|\frac{\cos\frac{x+a}{2}}{\sin\frac{x-a}{2}}=t\Rightarrow\frac{dx}{\sin^2\frac{x-a}{2}}=-\frac{2dt}{\cos a}\right|=-\frac{2}{\cos a}\int{t^{n-1}dt}=-\frac{2}{\cos a}\cdot\frac{t^n}{n}=-\frac{2}{n\cos a}\cdot\frac{\cos^n\frac{x+a}{2}}{\sin^n\frac{x-a}{2}}+C.$$

share|cite|improve this answer
Use double dollar signs (two before and two after the equation) renders the equation in "display mode", which uses larger font and more generous spacing. The result is much easier to read. – Austin Mohr Sep 5 '12 at 15:49

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.