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

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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.
 
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What is the derivative of sin? Do you remember substitutions?
 
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.
 
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?
 
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.
 
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
 
jedjj said:
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.
 
Last edited:
jedjj said:
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:
\intUdU

Can you integrate that? Then sub it back in with the same identities after you integrate
 
I think rocophysic's method is simpler, since it does not involve any substitutions.
 
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