Question

Find the area under the graph of f(x) = e-3ln(x) on the interval [1, 2].

Answer 1

is the function \[\LARGE f(x) = e^{-3\ln(x)}\] ??

Answer 2

Yes

Answer 3

Then what we're going to do is integrate the function from 1 to 2

Answer 4

you can make that function prettier before doing that whole integration thingy

Answer 5

You can simplify things a bit first
\[\LARGE f(x) = e^{-3\ln(x)}\]
\[\LARGE f(x) = e^{\ln(x^{-3})}\]
\[\LARGE f(x) = x^{-3}\]

Answer 6

\[\huge e^{-3\ln_e (x)} = e^{\ln_e(x^{-3})} = x^{-3} \] Does that make sense to you though?

Answer 7

I got 0.375

Answer 8

Yeah I understand how it's done.

Answer 9

\[\large \int_1^2 x^{-3}dx = \left.\frac{x^{-3+1}}{-3+1}\right]_{1}^2 = \left.\frac{x^{-2}}{-2}\right]_1^2=-\frac{1}{2}\left( \frac{1}{4} - 1\right) = \frac{3}{8} \qquad \checkmark\]