anonymous
  • anonymous
help me to integrate this : integration of (3 sin^2 x cos x) dx i confuse how to integrate this..
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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shubhamsrg
  • shubhamsrg
let sinx = t
Kainui
  • Kainui
Use "u"-substitution since you see a function and its derivative next to it.
anonymous
  • anonymous
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Kainui
  • Kainui
www.wolframalpha.com show me steps
anonymous
  • anonymous
^^shown
Australopithecus
  • Australopithecus
you have to use identities to solve this \[\int\limits_{}^{}3 \sin^2 x \cos x\]
Kainui
  • Kainui
...
Australopithecus
  • Australopithecus
\[3\int\limits_{}^{}\sin^2 x \cos xdx\] \[3\int\limits_{}^{}(1-\cos^{2}(x)) \cos xdx\]
Kainui
  • Kainui
No, sin^2(x)=u 1/2 du=cosxdx 3/2udu becomes your new integral.
Australopithecus
  • Australopithecus
f(x) = sin^2(x) f'(x) = cos(x)^2*sin(x)
anonymous
  • anonymous
Why not u = sinx --> du = cosxdx ??
anonymous
  • anonymous
= 3 ∫ u² du = ....
Australopithecus
  • Australopithecus
\[3(\int\limits_{}^{}\cos(x)dx - \int\limits_{}^{}\cos^2(x)dx)\]
anonymous
  • anonymous
It's crystal clear about the relation between sinx and cosxdx :)
Australopithecus
  • Australopithecus
remember for the future \[\cos^2(x) = \frac{1 + \cos(2x)}{2}\] so we have, \[3(\sin(x) + c - \int\limits_{}^{}\frac{1+\cos(2x)}{2}dx)\]
anonymous
  • anonymous
^done in 3rd reply
Australopithecus
  • Australopithecus
You can solve it my way too which is good practice for when you get integrals with trig functions to high powers
shubhamsrg
  • shubhamsrg
@Australopithecus its cos^3 x there and not square
Australopithecus
  • Australopithecus
oh sorry made a mistake lol
anonymous
  • anonymous
where the best solution? i still confuse.. hu2
Australopithecus
  • Australopithecus
it can still be solved with that method
shubhamsrg
  • shubhamsrg
and sinx= t ,u whatever, thats the easiest thing which you can do..as i pointed out in the very beginning ..
shubhamsrg
  • shubhamsrg
cos^3 x will still involve substitution ..
Australopithecus
  • Australopithecus
\[3(\sin(x) + c - \int\limits_{}^{}\cos(x)\frac{1 + \cos(2x)}{2}dx\]
Australopithecus
  • 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
Australopithecus
  • Australopithecus
I think I made a mistake again though meh
Kainui
  • Kainui
@Australopithecus It's like suggesting someone make a tunnel through a mountain when you can just use a teleporter.
Australopithecus
  • Australopithecus
It is still applicable and stop hassling me I'm rusty ha
anonymous
  • anonymous
so, the best solution? anyone?
anonymous
  • anonymous
3rd
anonymous
  • anonymous
@sha0403, to clarify, you should use any method provided except @Australopithecus'
anonymous
  • anonymous
ok thanks u all for helping me..i appreciate it.. =)

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