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:

 \displaystyle \inte10x dx = 1/10 x10x

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




\displaystyle \intxe10x = x * 1/10e10x  - \displaystyle \int1/10e10x dx


 = x /10e10x - 1/100e10x + c


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