Question

How do you find the definite integral of `2/(sqrt(x) *e^-sqrt(x))` from `[1, 4]`?

Answer 1

`4e^2-4e`

Explanation:

`I=int_1^4 2/(sqrtx*e^(-sqrtx))dx`

Simplify the negative exponent by bringing it to the numerator.

`I=2int_1^4e^sqrtx/sqrtxdx`

We will use the substitution `u=sqrtx`. This implies that `du=1/(2sqrtx)dx`. Don't forget to plug the bounds of `1` and `4` into `sqrtx`.

`I=4int_1^4e^sqrtx/(2sqrtx)dx=4int_1^2e^udu`

The integral `inte^udu=e^u+C`:

`I=4[e^u]_1^2=4(e^2-e^1)=4e^2-4e`