Question

Can someone help me with an integral question?

Answer 1

Suppose that \[\int\limits_{1}^{x^3}f(t)dt = x^2e^x\] for all x. Then f(8) is?

Answer 2

the answer is \[2e ^{2}/3\] but I don't know how to solve this question

Answer 3

try differentiating both sides by x, the right hand side should be fairly obvious, but the left hand side is a little more subtle with the limits depending on x.

Answer 4

I still don't get it... for the right side, it would be \[2xe^x + x^2e^x\] but how would you differentiate the left side if there is no actual equation to start with?

Answer 5

well, since only your upper limit depends on x, the function evaluated anywhere else is 0, so all we need concern ourselves with is the upper limit. the derivative in this case is the function evaluated at the upper limit, times the derivative of the upper limit \[\frac{d}{dx} \int_1^{x^3}f(t)dt=f(x^3)\frac{d ~x^3}{dx}\]

Answer 6

I don't understand how you derived the right side. I thought it was 2xe^x + x^2e^x?

Answer 7

i never did the right side, that was the left side evaluated, you still need to solve for f(t), or in this case f(x^3): at this point you should have something like: \[3x^2f(x^3) = 2xe^x+x^2e^x\]

Answer 8

Oh I get it. Thanks a lot! :)

Answer 9

glad I could help, that's actually a really neat question, I might do something similar for my integral of the day tomorrow.

Answer 10

Haha, it was from a past exam for my calculus finals that's coming up.