## anonymous 5 years ago integral from 0 to 6pi, 7theta^2(sin(1/12theta))dtheta

1. dumbcow

integration by parts pull out the constant 7 first u = x^2 dv = sin(x/12) du = 2xdx v = -12sin(x/12)

2. amistre64

$\int\limits_{0}^{6\pi} 7 \theta^2 \sin(\frac{1}{12} \theta) d \theta$ ??

3. amistre64

or is that (7t)^2 ??

4. anonymous

its right the first way u have it

5. dumbcow

= -12x^2cos(x/12) +24[integral xcos(x/12) dx] repeat integration by parts u = x dv = cos(x/12) du=dx v = 12sin(x/12)

6. amistre64

if i gotta repeat integrations; i just make a table to keep them organized

7. anonymous

if you express sin theta in eular's form then the integration will be easied

8. anonymous

yeh, but no one wants complex numbers in the answer geezzz

9. amistre64

v up ------------------------ u down | sin(t/12) ------------------------- + | t^2 | -cos(t/12) /12 ------------------------- - | 2t | -sin(t/12) / 144 ------------------------- + | t | cos(t/12)/ 144(12) ------------------------- 0 right?

10. amistre64

not to forget the 7 tho lol

11. amistre64

- 7t^2 cos(t/12) /12 + 14t sin(t/12) / 144 + 7t cos(t/12)/ 144(12) i think

12. dumbcow

amistre when you integrated the sin and cos, you multiplied the inside instead of dividing integral sin(ax) = -1/a*cos(ax) =7[-12x^2 cos(x/12) + 288x sin(x/12) + (24)(144)cos(x/12)]

13. amistre64

still rusty at integration by parts; thnx :)

14. dumbcow

no problem table is good idea though

15. amistre64

i seen the 12 and forgot it was a fraction :)

16. anonymous

thanks guys :)

17. anonymous

The derivative of $7 \left(-1728 \left(-2+\frac{t^2}{144}\right) \text{Cos}\left[\frac{t}{12}\right]+288 t \text{Sin}\left[\frac{t}{12}\right]\right)$is$7 \left( 288 \text{ Sin}\left[\frac{t}{12}\right]+144 \left(-2+\frac{t^2}{144}\right) \text{Sin}\left[\frac{t}{12}\right]\right)$simplified,$7 t^2 \text{Sin}\left[\frac{t}{12}\right]$ The truth be told, I don't know how they did it.

18. anonymous

can u find the answer to integral of e^(6x) cos(7x)

19. anonymous

hold on.

20. anonymous

$\int\limits e^{6 x} \text{Cos}[7 x]dx = \frac{1}{85} e^{6 x} (6 \text{Cos}[7 x]+7 \text{Sin}[7 x])+c$Does OK?

21. anonymous

yes, thank u so much :)

22. anonymous

You might consider buying a Student version of Mathematica 8 if you intend to pursue a scientific or math career. At least you can verify your answers.

23. anonymous