find the integral for xe^10x

for this type of question, integral by parts should be used . This involves using the formula for intregation by parts:

lets first break apart the x and e10x into two parts - "u" and "v"

where u = x.

however, we need to find the value of v. in order to do this, we can integrate dv/dx in order to get to v. 

the value of dv/dx is : e10x

 

u = x             dv/dx = e10x

 

as seen in the formula, you need to have a value for u, dv, v and du. 

therefore in order to get du you must differentiate u:

u = x   

du/dx = 1

du = 1dx = dx

du = dx

 

in order to get v you need to integrate dv/dx:

 e10x dx = 1/10 x10x

now that we have both parts, we can put this back into the formula. 

 

 

xe10x = x * 1/10e10x  - 1/10e10x dx

 

 = x /10e10x - 1/100e10x + c