Watch your integral.

[Leonhard]

This puzzle is slightly more trickier than the one I posted a moment ago. Usually when teaching young people the subtle mistakes you can make with integrals, I usually give them this puzzle after they've learned the rule of doing integration by parts (especially if they think they know what they're doing).

In highschool notation the rule for integration by parts is stated as



Where



and



Now lets do the following integral



this gives us



Lets subtract the integral from both sides and we're now left with

[klaus54]

I never noticed this and can't figure it out.

I would first reduce the integrand to 1 and get x + C

K54

[klaus54]

I never noticed this and can't figure it out.

I would first reduce the integrand to 1 and get x + C

K54

P.S. You can express 1 some other ways and get the same "contradiction". E.g., try it with integrand X*X^-1, x=0.

[Juvenal]

I never noticed this and can't figure it out.

I would first reduce the integrand to 1 and get x + C

K54

Using integration by parts, as specified:

f(x) = e^x
g(x) = e^-x

F(x) = e^x
g'(x)= -e^-x

So

F(x)g(x) = 1

But you're quite right to note the contradiction stems from the constant of integration.

[Boxing Pythagoras]

Lets subtract the integral from both sides and we're now left with

Ooooh, I dig this one. I'm putting this in my pocket for later-- perhaps an extra credit assignment for students.

[Leonhard]

I'm also kinda fond of it, can't remember where I've seen it though.