Question

integrate cos(theta)sin(theta)?

Answer 1

Are you familiar with u-substitution?

Answer 2

yes

Answer 3

That is what you will be doing here. Now, do you know how to go about it? Or do you want me to explain?

Answer 4

I'm rusty with it so please explain.

Answer 5

Well, basically what you are doing here is picking a term to substitute, so that you can use one of the basic integral rules that you were taught prior to this. Now, notice that you have \(sin(\theta)\) and \(cos(\theta)\).

You know from early on in calculus that the dervaitve of sine is cosine. Correct?

Answer 6

yes and the derivative of cosine is negative sine

Answer 7

So, pick a substitute! In this case, you are going to set \(u\) = sin(\(\theta\)).
Next, you are going to take the derivative of u, so tha you can get something to substitue with \(d\theta\), so that du = \(d\theta\)

Answer 8

ok im starting to remember

Answer 9

is it cos(theta)/du

Answer 10

\[\int\limits udu\] is what you will have when you use the following substitution.
u = \(sin\theta\)
du = \(\cos(\theta)\)

REMEMBER That you have \[\int\limits \sin(\theta) \sf\color{red}{cos(\theta )d\theta}\]

Answer 11

Are you following me? You basically made a substitution to make the integration easier for you.
Now, that you have \[\int udu\]

Answer 12

You can perform the basic integral rules that you previously learned. \(\large\frac{u^{n+1}}{n+1}\)

Answer 13

can you do the actual substitution part plz

Answer 14

\(\frac{u^2}{2}+C\) is your answer. But remember that your \(\sf\color{red}{original}\) fnction was in terms of \(\sf\color{red}{\theta}\), So you must change it back to what you had before, where u = \(sin\theta\).

So your FINAL answer is: \(\sf\color{red}{\frac{(sin(\theta))^2}{2}+C}\)

Answer 15

so where does the cosine belong when i plug sine back in

Answer 16

Nowhere. Your cosine was just what you used to substitute for \(du\)

Answer 17

It was merely the derivative of \(u\)! Because you cannot have two different terms, so you need a substitute for \(du\) also. If you didn't take a derivative, you would of had
\[\int\limits \sf\color{red}{u} \cos(\theta)d\theta\]

Answer 18

thanks man