Integrals + Trig: Solve sin(2x)/23+cos(x)^2 dx

[SciSteve]

I was reviewing my first calc class stuff before starting this second one and came across a problem that i can't seem to get, its the integral of sin(2x)/23+cos(x)^2 dx, i know most of the rules and thought i had it but the question asks to put in all trig functions in terms of cos which I can't seem to figure out how to do. Been awhile since I've done this stuff so sorry if its real easy and I am just missing something simple. thanks in advance.

 

[rock.freak667]

How about let $u=23+cos^2x$

 

[rocomath]

What does the double angle identity $$\sin{2x}$$ simplify to?

 

[HallsofIvy]

Is this a "differential equations" problem?

 

[SciSteve]

no i don't believe so, its an integral problems that i jus don't understand thought this would be the best place to put it.

 

[coomast]

What exactly is the question? Is it:

$$A=\int \frac{sin(2x)}{23+cos(x^2)}dx$$

or:

$$B=\int \left(\frac{sin(2x)}{23}+cos(x^2) \right)dx$$

or:

$$C=\int \frac{sin(2x)}{23+cos^2(x)}dx$$

or:

$$D=\int \left(\frac{sin(2x)}{23}+cos^2(x) \right)dx$$

It is unclear what you mean.

 

[Gib Z]

Almost certainly C. Though I get your point, careless notation is annoying.

 

[SciSteve]

it is the C one u posted IDK how to make it clearer writing all these functions n stuff out with a keyboard

 

[coomast]

This is a link where you can find the basics:

https://www.physicsforums.com/misc/howtolatex.pdf

If you click on a formula, the latex code pops up. No worries about the typing, you'll learn it. So the third one is the one you need to tackle. What have you got so far?

 

[Hootenanny]

it is the C one u posted IDK how to make it clearer writing all these functions n stuff out with a keyboard

In that case consider re-writing the denominator;

$$23+cos^2(x) = 23 + \frac{1}{2}\left(1 + \cos(2x)\right) = \frac{1}{2}\left(47+\cos(2x)\right)$$

Now take the derivative and compare with the numerator.