Problem with improper integrals

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The discussion revolves around evaluating the improper integral ∫xe^(-2x)dx from 0 to ∞. The initial attempt at a solution led to confusion regarding the limits and resulted in an incorrect value. After clarification, it was noted that the lower limit must be properly accounted for, leading to the correct evaluation. The final resolution indicated that the integral equals 1/4 after correcting the calculations. The participant acknowledged their mistake and expressed gratitude for the assistance.
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Homework Statement



∫xe^(-2x)dx from x = 0 to ∞

Homework Equations



-xe^(-2x)/2 - e^(-2x)/4 + C

The Attempt at a Solution



lim b→∞ -x/2e^(2b) - 1/4e^(2b) = 0

wolfram alpha says its 1/4 and I do not know why (it does not show steps)

Can you help me?
 
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You did not take the lower limit into account. Substitute x=0 and subtract.

ehild
 
You're right lol I'm dumb... anyway substituting the lower limit I get 3/4 and not 1/4 :/
-(-1/2 - 1/4) = 3/4
 
Thank you
 
just saw my mistake... 1/2 should be zero, that way I do get 1/4... thank you again
 

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