# Recursive Integral Simplification

#### Nihilist Comedian

$$\int{\sin^{n}(x)\cos^{m}(x)dx}$$
$$=\frac{\sin^{n+1}(x)\cos^{m-1}(x)}{n+1}+\frac{m-1}{n+1}\int{\sin^{n+2}(x)\cos^{m-2}(x)dx}$$

That was quite easy, but it's the simplification process following this that throws me. My answer is perfectly correct, but it is simplified in the answers (in my maths book) to the following form.

$$\frac{\sin^{n+1}(x)\cos^{m-1}(x)}{n+m}+\frac{m-1}{n+m}\int{\sin^{n}(x)\cos^{m-2}(x)dx}$$

The form that I had it in can be used to calculate integrals for specific values of n an m, though in an exam, I believe that I'd have to express it in simpler form to get full marks.

Thanks for the help.

Last edited by a moderator:
Related Introductory Physics Homework Help News on Phys.org

#### Nihilist Comedian

Sorry, forgot to put in the "n" and "m" in the original function.

Any help now?

#### arildno

Homework Helper
Gold Member
Dearly Missed
You have the following identity:
$$\int\sin^{n+2}x\cos^{m-2}xdx=\int\sin^{n}x(1-\cos^{2}x)\cos^{m-2}xdx$$

#### Nihilist Comedian

Thanks. It's really quite simple. I can't believe I missed that!

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving