Integrating Sin(\phi)Cos(\phi): A Step-By-Step Guide

[jedjj]

Homework Statement

I'm actually in the middle of a multivariable question, and I am stuck because I don't remember how to integrate $$sin(\phi)cos(\phi)$$ .

The Attempt at a Solution

I have an understanding of the material, but I can't remember how to integrate this. Someone please refresh my memory :) . I would appreciate some kind of step by step integration, so if this was on the test I would understand how to do it.

 

[Antineutron]

What is the derivative of sin? Do you remember substitutions?

 

[jedjj]

The derivative of sin(x) is cos(x), and I do remember substitutions, but I don't know what to substitute, because I can't remember any identities for sin or cos with a power of 1.

 

[Antineutron]

Do you remember substituting for U then finding the dU, which is the derivative of the U, then making substitutions to the original integral to change the integral interms of the variable U? Make U=sin(x), then what is dU?

 

[jedjj]

That is so weird. I didnt even think about that. Can you explain why that works, but the integral of sin(x) by itself is -cos(x)? (if your answer is 'it just is' that is perfectly fine). I thought of u substitution--I was thinking that it wouldn't work.

 

[rocomath]

maybe this will make it easier

$$sin{2x}=2sin{x}cos{x}$$

$$\int\sin{x}cos{x}dx$$

so

$$\frac{1}{2}\int\sin{2x}dx$$

 

[Antineutron]

That is so weird. I didnt even think about that. Can you explain why that works, but the integral of sin(x) by itself is -cos(x)? (if your answer is 'it just is' that is perfectly fine). I thought of u substitution--I was thinking that it wouldn't work.

messed up with my latex, still trying to get the hang of it.

 

[Antineutron]

That is so weird. I didnt even think about that. Can you explain why that works, but the integral of sin(x) by itself is -cos(x)? (if your answer is 'it just is' that is perfectly fine). I thought of u substitution--I was thinking that it wouldn't work.

Let U=sinx

then, dU=cosxdx

so if you substitute thes identities to the original equation:
$$\int$$UdU

Can you integrate that? Then sub it back in with the same identities after you integrate

 

[cristo]

I think rocophysic's method is simpler, since it does not involve any substitutions.