Integration?

Question

Hey differentiating (x^-4.y) gives (x^-4.y')-(4*x^-5.y) (using the product rule), i have been struggling for ages to integrate the latter expression but can't show it equals x^4.y. Can anyone help? I've been trying integration by parts i'm presuming this is the right method and i'm just making a silly mistake.

Answer 1

I'll use * for multiplication. (I think thats what was throwing off the other answerer.)

S = ∫ x^(-4) * y' dx - ∫ 4*x^(-5) * y dx

Use integration by parts for just the first integral
set u = x^(-4) and v' = y'
so u' = -4x^(-5) and v = y

S = uv - ∫ vu' dx - ∫ 4*x^(-5) * y dx
= x^(-4) * y - ∫ [-4x^(-5) * y] dx - ∫ 4*x^(-5) * y dx
= x^(-4) * y + ∫ 4*x^(-5) * y dx - ∫ 4*x^(-5) * y dx
= x^(-4) * y

{you also could have done it using integration by parts on just the second integral with u=y and v'=4*x^(-5). }

Answer 2

Some functions cannot be integrated.

In a blue bookk exam I strugled for several pages trying to integrate a function and could not do it. The professor wrote "Heroic Effort" and gave me full credit. Then he continued "All you had t o do do was realize that in the previous problem you correctly differentiated a function that this problem is now asking you to integrate the derivative you obtained in the previous problem." Man , I felt foolish.

He later told me the function could not be integrated by conventional methods. Pperhaps your problem is the same.

Answer 3

Can you restate your question? For one thing, I can't make sense out of the "5.y" that you write, because usually all the variables in a function to integrate are all x. Can you maybe clarify?