Understanding Integral Calculus: Solving ∫cos^3xdx with Step-by-Step Guidance

[guitarphysics]

Homework Statement

∫cos^3xdx

Homework Equations

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?

 

[dextercioby]

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

 

[Mandelbroth]

Homework Statement

∫cos^3xdx

Homework Equations

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?

\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.

 

[DryRun]

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,
$$\int cos^3 \theta\,.d\theta= \frac{1}{4} \int (\cos 3\theta + 3\cos \theta)\,.d\theta$$

 

[haruspex]

\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.

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

 

[HallsofIvy]

Yes, that's what madelbroth meant.

 

[haruspex]

Yes, that's what madelbroth meant.

I know, just saying you don't have to go through a formal substitution, adjusting limits.