(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that

[tex]

\int cos^mxdx = \frac{cos^{m-1}x sinx}{m} + \frac{m-1}{m}\int cos^{m-2}xdx

[/tex]

2. Relevant equations

[tex]

\int f(x)g'(x) = f(x)g(x) - \int g(x)f'(x) dx

[/tex]

3. The attempt at a solution

Going through any of the integrals provides constants that seem to be a problem in proving the above.

Starting from the left side and applying the integral product rule (see 2. -- not sure how the rule is called in English!) we get:

[tex]

\int cos^mxdx = \int cos^{m-1}x \cdot cosx dx

[/tex]

With [tex]f(x) = cos^{m-1}x[/tex], [tex]g'(x) = cosx[/tex], [tex]g(x) = sinx[/tex], [tex]f'(x) = (m-1)cos^{m-2}x[/tex] we get:

[tex]

cos^{m-1}x \cdot sinx - \int sinx(m-1)cos^{m-2}x dx = cos^{m-1}x \cdot sinx - (m-1)\int cos^{m-2}x \cdot sinx dx

[/tex]

With [tex]f(x) = cos^{m-2}x[/tex], [tex]g'(x) = sinx[/tex], [tex]g(x) = -cosx[/tex], [tex]f'(x) = (m-2)cos^{m-3}x[/tex] we get:

[tex]

cos^{m-1}x \cdot sinx - (m-1)\left( cos^{m-2}x (-cosx) - \int (-cosx)(m-2)cos^{m-3}x dx \right) =

[/tex]

[tex]

cos^{m-1}x \cdot sinx - (m-1)\left( -cos^{m-1}x + (m-2)\int cos^{m-2}x dx \right) =

[/tex]

Any ideas on how to continue?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Proof of equation with integrals

**Physics Forums | Science Articles, Homework Help, Discussion**