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

• SciSteve
In summary: So this integral is equivalent to\int \frac{d}{dx}[23+cos^2(x)]=-\frac{1}{2}\left(47+\cos(2x)\right)=-\frac{23}{4}
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.

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

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

Is this a "differential equations" problem?

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.

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.

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

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

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?

SciSteve said:
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.

## 1. What is an integral?

An integral is a mathematical concept that represents the area under a curve or the accumulation of a quantity over a certain interval. It is denoted by the symbol ∫.

## 2. How do you solve an integral?

To solve an integral, you need to use integration techniques such as substitution, integration by parts, or trigonometric identities. You also need to have a good understanding of the fundamental theorem of calculus.

## 3. What is the purpose of using trigonometric functions in integrals?

Trigonometric functions, such as sine and cosine, are often used in integrals because they can help simplify the integration process. They also have many real-world applications in fields such as physics and engineering.

## 4. What does the notation "dx" mean in an integral?

The "dx" in an integral represents the variable of integration, which is the independent variable in the function being integrated. It is used to indicate the variable with respect to which the integral is being evaluated.

## 5. How do you solve the integral of sin(2x)/23+cos(x)^2?

To solve this integral, you can use the trigonometric identity cos(2x) = 1 - 2sin^2(x). This will allow you to rewrite the integral as ∫(sin(2x)/23 + 1 - 2sin^2(x))dx. You can then use the linearity property of integrals to split it into two separate integrals and use the power rule and trigonometric identities to solve each one.

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