Question

Integration by substitution

Answer 1

where is question?

Answer 2

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

Answer 3

is it supposed to be cos instead of sin?

Answer 4

keep your derivatives in mind, what happens if you derive -cos(x)?

Answer 5

sin

Answer 6

exactly.

Answer 7

oh i just checked the answer sheet -_- it says cos...

Answer 8

1/2 (cos(2x)) + C

Answer 9

hmm, well that's not the answer in my opinion, because if you derive that you get:

\[\Large - \frac{1}{2} \sin(2x) \cdot 2 = - \sin(2x) \]

Answer 10

oh... i didn't get that at all. idk, the given answer is 1/2cos(2x) + C

Answer 11

is my u sub wrong?

Answer 12

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

\[F(x)\prime =f(x) \]

Answer 13

OHHHHHH snap I get it! lol

Answer 14

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.

Answer 15

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?

Answer 16

oh the constant part i have no prob with hahaha general integrals = add or subtract c lol

Answer 17

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.

Answer 18

i'm not sure if that is sarcasm because i was tempted to give you the definition lolll but thanks!

Answer 19

No, not sarcasm, just discouragement. I don't recommend that usage. It just isn't generally in use.

Answer 20

oh here it is. as long as we know the answer and the concept, it's all good.

Answer 21

exactly.

Answer 22

And perhaps that we managed to learn something else along the way. Good work.