Integration by substitution

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Integration by substitution

Mathematics
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general integral with the function sin(2x) dx My work: u = 2x du/dx = 2 du = 2dx 1/2 integral (2sin(2x)dx) 1/2 integral (sinu * du) 1/2sin(2x) + C but that's not right i believe.. lol
is it supposed to be cos instead of sin?

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keep your derivatives in mind, what happens if you derive -cos(x)?
sin
exactly.
oh i just checked the answer sheet -_- it says cos...
1/2 (cos(2x)) + C
hmm, well that's not the answer in my opinion, because if you derive that you get: \[\Large - \frac{1}{2} \sin(2x) \cdot 2 = - \sin(2x) \]
oh... i didn't get that at all. idk, the given answer is 1/2cos(2x) + C
is my u sub wrong?
No your substitution is perfect, it's more the way you integrated, you kept the sin function, which shouldn't be the case when you integrate, because when you differentiate you want to obtain the integrand again \[F(x)\prime =f(x) \]
OHHHHHH snap I get it! lol
Seriously? Why would you EVER use substitution for a mere constant? It's just craziness. Speculate \(\int \sin(2x)\;dx = -\cos(2x)\;+\;C\) Check \(\dfrac{d}{dx}(-\cos(2x)) = 2\cdot \sin(2x)\) -- Oops, we missed a constant. Solve \(\int \sin(2x)\;dx = -\dfrac{1}{2}\cos(2x)\;+\;C\) -- Done. On the other hand: \(\int \sin(2x)\;dx = \int 2\sin(x)\cos(x)\;dx\) -- Now, THERE'S a candidate for Substitution.
because I've been doing exponential functions all day -_- so i kept that in mind that i shouldn't change it. But just to make sure... i should integrate the function after i derive the u correct?
oh the constant part i have no prob with hahaha general integrals = add or subtract c lol
I wish I could tell what "derive" means. I have not found it acceptable to use it as the verb form for finding a derivative.
i'm not sure if that is sarcasm because i was tempted to give you the definition lolll but thanks!
No, not sarcasm, just discouragement. I don't recommend that usage. It just isn't generally in use.
oh here it is. as long as we know the answer and the concept, it's all good.
exactly.
And perhaps that we managed to learn something else along the way. Good work.

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