6.1.2 Antiderivatives Involving Exponential and
         Logarithmic Functions

Example 4

Find an antiderivative for `e^(5t).`

Solution   Since `d/(dt)e^t=e^t,` we know by the Chain Rule that `d/(dt)e^(5t)=5e^(5t).` Thus an antiderivative for `e^(5t)` is `1/5e^(5t).`

Checkpoint 4

Example 5

Find an antiderivative for `1/(3x+1).`

Solution   The reciprocal function is the one exception to our general rule for finding antiderivatives of power functions:

`x^(r+1)/(r+1)` is an antiderivative for `x^r,` except when `r=-1.`

However, we know an antiderivative for `1/x`:

Thus, by the Chain Rule,

Using the Constant Multiple Rule we see that

So, `1/3 ln(3x+1)` is an antiderivative for `1/(3x+1).`

Checkpoint 5

Next we note that the Sum Rule for differentiation enables us to calculate antitderivatives term by term.

Example 6

Find an antiderivative for `3x^5+sin(x/2)-e^(-x).`

Solution   Taking each of the terms separately, we calculate the following:

• An antiderivative for `3x^5` is `1/2 x^6.`

• An antiderivative for `sin(x/2)` is `-2 cos(x/2).`

• An antiderivative for ` -e^(-x)` is `  e^(-x).`

Putting this together, we have the antiderivative that we want:

Checkpoint 6

Checkpoint 7