anonymous 3 years ago Integration by substitution

1. amoodarya

where is question?

2. anonymous

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

3. anonymous

is it supposed to be cos instead of sin?

4. anonymous

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

5. anonymous

sin

6. anonymous

exactly.

7. anonymous

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

8. anonymous

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

9. anonymous

hmm, well that's not the answer in my opinion, because if you derive that you get: $\Large - \frac{1}{2} \sin(2x) \cdot 2 = - \sin(2x)$

10. anonymous

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

11. anonymous

is my u sub wrong?

12. anonymous

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 $F(x)\prime =f(x)$

13. anonymous

OHHHHHH snap I get it! lol

14. tkhunny

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.

15. anonymous

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?

16. anonymous

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

17. tkhunny

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.

18. anonymous

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

19. tkhunny

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

20. anonymous

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

21. anonymous

exactly.

22. tkhunny

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