1. ## help with integration/differentiation

Hi. I need help with a Maths number.

∫ f(x) dx = (b^2) (e^b) - e

where the upper limit of the integral is b and the lower limit is 1. Also, b > 0

Find an expression for f(x) for all x>0.

I know I have to differentiate the b(sqaure) * e(to the power of b) - e, but I don't really know how to do it since there are no x in the expression. I know I should be using the limits of the integrals as well but I don't know how :/
Help! Thanks x

2. Differentiate

$\displaystyle \displaystyle\int_{1}^{b}f(x)\,dx=b^{2}e^{b}-e$

with respect to $\displaystyle b.$ Use the Fundamental Theorem of the Calculus to differentiate the LHS.

3. I can't do that. I need to differentiate it w.r.t x not b. I have to use f(b) - f(1) =(b^2) (e^b) - e and then try and find the expression for x. I just don't know how to proceed.

4. Technically speaking, if you differentiated both sides, right now, w.r.t. x, you will get zero on both sides. That will give you no information whatsoever. You can rewrite the equation if you like:

$\displaystyle \displaystyle\int_{1}^{x}f(t)\,dt=x^{2}e^{x}-e.$

Now differentiate w.r.t. x.

5. when I differentiate it, I get

f(b) = (x^2)*(e^b) + (2b)*(e^b)

so the expression for f(x) should be (x^2)*(e^x) + (2x)*(e^x)?

but when I integrate it to check if it is equal to (b^2) (e^b) - e, I can't find the solution.

6. Integrate the function from 1 to b. What do you get?

7. ohh ok..I managed to do it..thanks!

8. You're welcome. Have a good one!