sha0403 2 years ago help me to integrate this : integration of (3 sin^2 x cos x) dx i confuse how to integrate this..

1. shubhamsrg

let sinx = t

2. Kainui

Use "u"-substitution since you see a function and its derivative next to it.

3. stgreen

4. Kainui

www.wolframalpha.com show me steps

5. stgreen

^^shown

6. Australopithecus

you have to use identities to solve this $\int\limits_{}^{}3 \sin^2 x \cos x$

7. Kainui

...

8. Australopithecus

$3\int\limits_{}^{}\sin^2 x \cos xdx$ $3\int\limits_{}^{}(1-\cos^{2}(x)) \cos xdx$

9. Kainui

No, sin^2(x)=u 1/2 du=cosxdx 3/2udu becomes your new integral.

10. Australopithecus

f(x) = sin^2(x) f'(x) = cos(x)^2*sin(x)

11. Chlorophyll

Why not u = sinx --> du = cosxdx ??

12. Chlorophyll

= 3 ∫ u² du = ....

13. Australopithecus

$3(\int\limits_{}^{}\cos(x)dx - \int\limits_{}^{}\cos^2(x)dx)$

14. Chlorophyll

It's crystal clear about the relation between sinx and cosxdx :)

15. Australopithecus

remember for the future $\cos^2(x) = \frac{1 + \cos(2x)}{2}$ so we have, $3(\sin(x) + c - \int\limits_{}^{}\frac{1+\cos(2x)}{2}dx)$

16. stgreen

17. Australopithecus

You can solve it my way too which is good practice for when you get integrals with trig functions to high powers

18. shubhamsrg

@Australopithecus its cos^3 x there and not square

19. Australopithecus

oh sorry made a mistake lol

20. sha0403

where the best solution? i still confuse.. hu2

21. Australopithecus

it can still be solved with that method

22. shubhamsrg

and sinx= t ,u whatever, thats the easiest thing which you can do..as i pointed out in the very beginning ..

23. shubhamsrg

cos^3 x will still involve substitution ..

24. Australopithecus

$3(\sin(x) + c - \int\limits_{}^{}\cos(x)\frac{1 + \cos(2x)}{2}dx$

25. Australopithecus

I'm not arguing that substitution will be involved in my solution but it still works for solving this integral it is just the long way

26. Australopithecus

I think I made a mistake again though meh

27. Kainui

@Australopithecus It's like suggesting someone make a tunnel through a mountain when you can just use a teleporter.

28. Australopithecus

It is still applicable and stop hassling me I'm rusty ha

29. sha0403

so, the best solution? anyone?

30. stgreen

3rd

31. Mathmuse

@sha0403, to clarify, you should use any method provided except @Australopithecus'

32. sha0403

ok thanks u all for helping me..i appreciate it.. =)