Question

integrate sin³ 4x dx

Answer 1

whenever you have an odd degree trig like that, its prolly a good idea to split it apart and swap about some identities

Answer 2

or there is a formula that is a bit lengthy that you can try to recall.

Answer 3

simply let cos4x =t

see if it helps ?

Answer 4

no can't use the substitution method, because the derivative of the function is not available, what you need to do is expand the trig function using trig identities

Answer 5

Of course derivative of the function is available! What are you saying! :O

Answer 6

you are saying t=cos(4x), then dt/dx=-4sin(4x), y=int t^4 so where are you leading this problem into more mess...

Answer 7

you should assume t=4x i think this is what you meant...

Answer 8

if this is the case then you can solve it using the integral formula for powers of sin or cos

Answer 9

or we split that and use integration by parts after applying trig identities which will lead to the derivation of the reduction integral formula of trig to the power functions...

Answer 10

cos4x = t
-4 sin4x dx = dt

our ques is

sin^3 (4x) dx= sin(4x)* sin^2(4x) dx = ((sin4x)dx)*(1-cos^2(4x)dx)


= (1-t^2) dt/(-4)


what part of this seems complicated to you?

Answer 11

so you are integrating 1-t^2/(-4)

Answer 12

integrate that you won't get the right answer...

Answer 13

-_-

Answer 14

Any reason why not?

Answer 15

because when you integrate the function it contradicts with the reduction forumal answer

Answer 16

even by using trig identity it can't be reduced to what you are kindly suggesting...

Answer 17

i don't mind if there are different ways to solve the same problem but we should be able to justify our answers...

Answer 18

split that trig function and using integration by parts, no way by any means that would be the same as 1-t^2

Answer 19

let me give you a slang hint, why would we need a reduction formula, if it could be just solve by substituting the whole trig function...

Answer 20

You are making absurd hypothesis.

How does it matter which path we chose when the final destination is the same!

Of course the 2 answers will match whatever the path taken be, given that we have done it legitimately .
Well mine is surely correct, check yours.

Answer 21

this is my point i said they don't end up to the same results even by any trig expansion or simplification...

Answer 22

the only way you can use u=cos(4x), after you use trig identity, not the other way round...

Answer 23

you have to expand the rig identity in the form S^(n+1) then use the substitution method u=cos(4x)...

Answer 24

so you need to make one substitution in order to apply u=cos(t),

Answer 25

where t = 4x, otherwise it won't be valid... logically, unless you want to say you know it by head, but the student does not know

Answer 26

now with those two substitution we can match with the reduction formula with no problem

Answer 27

what i would use to save time on this would be Euler's identity.. for complex exponential and trig functions...

Answer 28

in other words do not jump steps...that will reduce your mark