# Need help on integral problem

1. Dec 21, 2012

### guitarphysics

1. The problem statement, all variables and given/known data
∫cos^3xdx

2. Relevant equations

3. The attempt at a solution
I started teaching myself integral calculus yesterday and I had no idea what to do for this problem... I tried splitting cos^3x into cos^2x(cosx) but that didn't work. How do you solve this?

2. Dec 21, 2012

### dextercioby

You need to split it the way you did it and use trigonometry. $\cos^2 x = 1-\sin^2 x$ plus substitution.

3. Dec 21, 2012

### Mandelbroth

$\displaystyle \int cos^n {x} \ dx = \frac{sin{x} \ cos^{n-1}{x}}{n} + \frac{n-1}{n} \int cos^{n-2}x \ dx$.

Alternatively, $\displaystyle \int cos^3 {x} \ dx = \int cos^2 x \ cosx \ dx = \int (1-sin^2 x) cos x \ dx$. From here, you could use substitution.

4. Dec 21, 2012

### sharks

Do you know about the triple angle formulas? Maybe you've come across it in trigonometry:
$$\cos 3\theta =4\cos^3 \theta - 3\cos \theta$$
Re-arranging:
$$cos^3 \theta = \frac{1}{4} (\cos 3\theta + 3\cos \theta)$$
Therefore,
$$\int cos^3 \theta\,.d\theta= \frac{1}{4} \int (\cos 3\theta + 3\cos \theta)\,.d\theta$$

5. Dec 21, 2012

### haruspex

... or just use cos x dx = d sin x.

6. Dec 22, 2012

### HallsofIvy

Staff Emeritus