Question

Find the integral of x^5 cos (x^3) dx

Answer 1

use integration by parts !

Answer 2

ilate rule !

Answer 3

I know, but I'm having troule on the antiderivative of v

Answer 4

x^3*(x^2cos(x^3)) and integrate by parts

Answer 5

u=x^5
v=cos(x^3)

Answer 6

How will that v work?

Answer 7

d/dx(sinx)=cosx

Answer 8

antiderivative of cos x =sinx

Answer 9

antiderivative =integral.

Answer 10

for my u i picked x^3 and for my du= 3x^2

Answer 11

u=x^5 !

Answer 12

why pick u= x^5?

Answer 13

because we haVE TO FIND the integral of x^5 cos (x^3) .

Answer 14

u=x^5
v=cos(x^3)

Answer 15

u=x^3,

dv=x^2 cos x^3 dx

I think you were on the right track.
Not sure what guru is talking about :o

Answer 16

^yep the reason you do it that way is because you can easily use u-sub on x^2 cos(x^3)

Answer 17

You might have to integrate it by parts more than once though, i haven't worked through it yet, but sometimes in these cases you need to.

Answer 18

how do I find v?

Answer 19

You'll need to make a substitution within your v, it's a little tricky.

Let m=x^3 or something like that. ( We don't want to use u again).

Answer 20

can you draw that? I'm a visual learner.

Answer 21

v = int x^2 cos x^3 dx

u = x^3 ----> (1/3)du = x^2 dx

v = int cos u (1/3)du



Unfortunately the drawing tool isn't working right now :(
Otherwise I would..

Answer 22

ok, I'm still confused but thanks for your help!

Answer 23

I did a sneaky little trick there.
I multiplied by 3 and left a factor of 1/3 outside of the integral.
Our dv will change to v a little easier that way.

I may have skipped a few steps in there :o lemme know if it's not clear.

Answer 24

where did the 1/3 come from?

Answer 25

to me, I let u = x^6 --> x^3 = sqrt (u)
and du = 6x^5 dx
so, the integral = 1/6 integral cos (sqrt u) du

Answer 26

^ that might be just as awkward to integrate

Answer 27

hehehe... may be!! http://www.wolframalpha.com/input/?i=integral+%28cos+%28sqrt+%28u%29%29%29du

Answer 28

http://www3.wolframalpha.com/Calculate/MSP/MSP7051ga2a41igd48409400003ii2a968aed0fh3h?MSPStoreType=image/png&s=45&w=458&h=648

maybe it's easier than the other method, but i haven't tried the other yet