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

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

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