Integration of Parts Calculus help

[SEG9585]

Hey all--
I had an Integration of Parts quiz today and got stuck on a few problems-- was wondering if you could explain the steps involved in solving these integrals:

int( (sin(3x))^3 * (cos(3x))^3 dx)

and

int( (tan(4x))^4) dx)

thanks!

 

[cookiemonster]

$$\int \sin^3{3x} \cos^3{3x}\,dx$$

First things first, get rid of the 3x's with a u substitution. It's easy to see that all it does is change the solution by a factor of 1 over 3. So we have,

$$\frac{1}{3}\int \sin^3{u} \cos^3{u}\,du$$

If we had only 1 sine term or only 1 cosine term, we'd be gold. Problem solved. But we got 2 too many. So let's get rid of them!

$$\sin^2{x} + \cos^2{x} = 1$$

Use this to turn the integral into

$$\frac{1}{3}\int \sin^3{u} (1 - \sin^2{u}) \cos{u}\,du$$

which is easily separated and solved by substitution.

Have another shot at the second one, keeping in mind that

$$\frac{d}{dx}\tan{x} = \sec^2{x}$$

cookiemonster

 

[NateTG]

Where did you get $$\cos^3=1-\sin^2$$
I would probably use the half angle formulas:
$$\sin(2x)=2\sin(x)\cos(x)$$
so
$$\sin(x)\cos(x)=\frac{\sin(2x)}{2}$$
so
$$\sin(3x)\cos(3x)=\frac{\sin(6x)}{2}$$

Now you've got:
$$\int\sin^3(3x)\cos^3(3x)dx$$
$$\int(\sin(3x)\cos(3x))^3dx$$
$$\frac{1}{8}\int\sin^3(6x)dx$$

Now
$$\sin(3x)=3\sin(x)-4\sin^3(x)$$
so
$$\sin^3(6x)=\frac{3\sin(6x)-\sin(18x)}{4}$$
so
$$\frac{1}{32}\int3\sin(6x)-\sin(18x)dx$$
so
$$\int\sin^3(3x)\cos^3(3x)dx=\frac{\cos(18x)}{576}-\frac{\cos(6x)}{192}+C$$

Oh, by parts...

 

[cookiemonster]

I didn't. I only took two of the cosines and I left the third for the u substitution.

u = sinx
du = cosxdx

It's used in the du.

Edit: By parts?

cookiemonster