Question

How do you find the definite integral for: `e^(5x) dx` for the intervals `[0, 1]`?

Answer 1

`(e^5-1)/5`

Explanation:

We want to find:

`int_0^1e^(5x)dx`

Our goal for integration should be to get this integral into the pattern:

`inte^udu=e^u+C`

Thus, we substitute and let `u=5x`, so that `(du)/dx=5` and `du=5dx`.

To have our `du=5dx` value inside the initial integral, we will have to multiply the interior of the integral by `5`. Balance this by multiplying the exterior by `1/5`.

`=1/5int_0^1e^(5x)*5dx`

We now see that this will fit the `inte^udu` mold. However, be careful when substituting these in--since the integrand has changed from `dx` to `du`, we will have to change the bounds of integration as well.

Do this by plugging the current bounds of `0` and `1` into the equation for `u`, `u=5x`.

`u(0)=5(0)=0`
`u(1)=5(1)=5`

Thus,

`1/5int_0^1e^(5x)*5dx=1/5int_0^5e^udu`

We can now evaluate the integral from `0` to `5`:

`=1/5(e^u)]_0^5=1/5(e^5-e^0)=(e^5-1)/5`