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# How to integrate this equation? watch

1. Hi,

This is question 6 on my exercise sheet which you would assume to be easier than the other 10 questions after it. However I can't do it, I get repetitive motion when doing it in parts with substitution. What sort of method works?

Integrate(2*x*sin(x)*exp(x)) with respect to x of course.

Good luck to anyone who dares. I personally think that the question might be easier than it appears to be, just I am using the wrong method.

2. (Original post by byb3)
Hi,

This is question 6 on my exercise sheet which you would assume to be easier than the other 10 questions after it. However I can't do it, I get repetitive motion when doing it in parts with substitution. What sort of method works?

Integrate(2*x*sin(x)*exp(x)) with respect to x of course.

Good luck to anyone who dares. I personally think that the question might be easier than it appears to be, just I am using the wrong method.

What is exp(x)? I have only done up to P3.
3. (Original post by Ralfskini)
What is exp(x)? I have only done up to P3.
exp(x) is just e^x sorry
4. (Original post by byb3)
exp(x) is just e^x sorry

Thank.
5. Just don't - you only live once.
6. ralfskini check ur private messages
7. I get 2x + cosx*e^x + c.
8. (Original post by Seer)
ralfskini check ur private messages

Ive replied. Not sure if it sent.
9. Maple gives the answer as:

2*(-1/2*x+1/2)*exp(x)*cos(x)+x*sin(x)*exp(x)

= (1-x)*exp(x)*cos(x)+x*sin(x)*exp(x)

How it gets there is anyones idea.
10. (Original post by byb3)

2*(-1/2*x+1/2)*exp(x)*cos(x)+x*sin(x)*exp(x)

How it gets there is anyones idea.

I did a long integration by parts.
11. (Original post by Ralfskini)
I did a long integration by parts.
I think it is a mixture of that and substitution when needed.
12. I rushed it through. I can see the errors all over the place.
13. have you tried writing sin(x) in exponential form?
14. (Original post by elpaw)
have you tried writing sin(x) in exponential form?
sin(x) = exp(log(sin(x))) ?
15. First, do the following integrals (all with respect to x) (do them by two applications of parts, after which you will get the integral in terms of itself, which then after re-arranging will yield the following):

Code:
```I_1 = Integral(e^x*cos(x)) = (1/2)*e^x*(sin(x) + cos(x)) + A
I_2 = Integral(e^x*sin(x)) = (1/2)*e^x*(sin(x) - cos(x)) + B```
We want to find:

Code:
`(1/2)*I = Integral(x*sin(x)*e^x)`
Now attack this by parts. Set u = x, dv/dx = e^x*sin(x), thus du/dx = 1 and v = I_2 = (1/2)*e^x*(sin(x) - cos(x)). Then we have:

Code:
```(1/2)*I = (1/2)*x*e^x*(sin(x) - cos(x)) - (1/2)*I_2 + (1/2)*I_1 + C
I = x*e^x*(sin(x) - cos(x)) - I_2 + I_1 + D
I = x*e^x*(sin(x) - cos(x)) - (1/2)*e^x*(sin(x) - cos(x)) + (1/2)*e^x*(sin(x) + cos(x)) + E
I = x*e^x*(sin(x) - cos(x)) + e^x*cos(x) + E```
I hope this helps,
16. (Original post by rahaydenuk)
First, do the following integrals (all with respect to x) (do them by two applications of parts, after which you will get the integral in terms of itself, which then after re-arranging will yield the following):

Code:
```I_1 = Integral(e^x*cos(x)) = (1/2)*e^x*(sin(x) + cos(x))
I_2 = Integral(e^x*sin(x)) = (1/2)*e^x*(sin(x) - cos(x))```
We want to find:

Code:
`(1/2)*I = Integral(x*sin(x)*e^x)`
Now attack this by parts. Set u = x, dv/dx = e^x*sin(x), thus du/dx = 1 and v = I_2 = (1/2)*e^x*(sin(x) - cos(x)). Then we have:

Code:
```(1/2)*I = (1/2)*x*e^x*(sin(x) - cos(x)) - (1/2)*I_2 + (1/2)*I_1 + C
I = x*e^x*(sin(x) - cos(x)) - I_2 + I_1 + D
I = x*e^x*(sin(x) - cos(x)) - (1/2)*e^x*(sin(x) - cos(x)) + (1/2)*e^x*(sin(x) + cos(x)) + D
I = x*e^x*(sin(x) - cos(x)) + e^x*cos(x) + D```
I hope this helps,

Oh my god ... im glad i int doin maths at uni
17. (Original post by byb3)
sin(x) = exp(log(sin(x))) ?
no, sin(x) = (e^ix - e^-ix)/2i
18. I got:
Code:
`( xsinx - xcosx + cosx )e^x + C`
It took bloody ages... I just used integration by parts loads of times.
19. (Original post by mikesgt2)
I got:
Code:
`( xsinx - xcosx + cosx )e^x + C`
It took bloody ages... I just used integration by parts loads of times.
eeeek....i got (by parts):
Code:
`(2/3)(xsinx + (1 - x)cosx)e^x) + c`
20. (Original post by elpaw)
have you tried writing sin(x) in exponential form?
pretty pointless.

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