# [SOLVED] Trig Integrals

• Feb 5th 2009, 07:50 PM
mollymcf2009
[SOLVED] Trig Integrals
Hey everyone!

Integrate.

$\displaystyle \int 59 cos^2 x \cdot sin (2x) dx$

Here is what I have done so far:

$\displaystyle 59 \int (\frac{1}{2}(1 + cos 2x) \cdot (2 sinx cosx) dx$

Are those the correct identities to substitute in there?

Also need some help to go on to the next step of this.

Thanks y'all!
• Feb 5th 2009, 08:13 PM
Soroban
Hello, mollymcf2009!

Quote:

Integrate: .$\displaystyle 59\int \cos^2\!x\sin2x\, dx$

Here is what I have done so far: .$\displaystyle 59 \int \left(\frac{1 + \cos 2x}{2}\right)(2\sin x\cos x)\,dx$
Don't change both of them.

There are a few ways to approach this problem.
Here's one of them . . .

We have: .$\displaystyle 59\int\cos^2\!x(2\sin x\cos x)\,dx \;=\;118\int\cos^3\!x(\sin x\,dx)$

Let: $\displaystyle u \,=\, \cos x \quad\Rightarrow\quad du \,=\, -\sin x\,dx \quad\Rightarrow\quad \sin x\,dx \,=\,-du$

Substitute: .$\displaystyle 118\int u^3(-du) \;\;=\;\;-118\int u^3\,du \;\;=\;\;-118\,\frac{u^4}{4} + C \;\;=\;\;-\frac{59}{2}u^4 + C$

Back-substitute: .$\displaystyle -\frac{59}{2}\cos^4\!x + C$