Evaluate the integral `int(p^2)(lnp)dp`

Expert Answers
justaguide eNotes educator| Certified Educator

The integral `int p^2*ln(p) dp` has to be determined.

Let `ln(p) = y => p = e^y` , `p*dy = dp`

`int p^2*ln(p) dp`

=> `int e^(2y)*y*e^y dy`

=> `int e^(3y)*y*dy`

Use integration by parts, let u = y and v = e^(3y)

=> `y*e^(3y) - int e^(3y) dy`

=> `y*e^(3y) - e^(3y)/3`

substitute y = ln(p)

=> `ln(p)*e^(ln p^3) - (e^(lnp^3))/3`

=> `ln(p)*p^3 - p^3/3`

The integral `int p^2*ln(p) dp = ln(p)*p^3 - p^3/3 + C`

user6978788 | Student

Actually, the answer should be `1/3` x(what you got there)

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