Efficient Integration of x(e^(-x-theta)): Tips and Solutions

[confused88]

Homework Statement

Can someone please help me with integration of x(e^(-x-theta)) Thank you. My working is shown below

The Attempt at a Solution

u = x, du = 1 dx, dv = e^(-x-theta)) v = e^-x+theta+1 / theta + 1

the rest can be seen in the attachment. Can you please check it out, it would be greatly appreciated.Thanks

 

[JG89]

The attachment is pending approval so I can't see it.

Use the substitution u = -x - \theta \Rightarrow x = -(u+\theta) \Rightarrow dx = -du. So our integral is now \int (u + theta)e^u du.

Now use integration by parts.

 

[confused88]

ohhh wow thanks a lot!

 

[HallsofIvy]

The first thing I would do is write e^{-x- \theta}= e^{-x}e^{-\theta} and factor out the e^{-\theta}. Your integral is just e^{-\theta}\int xe^{-x}dx which can be done by integration by parts.