Integrating with Letters Examples

Find  where a is a constant greater than zero.

We need an antiderivative of x^a. This looks like it came from the power rule, so the simplest antiderivative is

Now we can use that in the FTC:

This is as nice as the answer gets.

Let a, b, c be nonzero constants. Find .

The simplest antiderivative of ax² + bx + c is

We can use this in the FTC, plugging the limits of integration into the variable of integration, which is x (not a or b or c):

At this point, we collect like terms so we can get all the a terms together, all the b terms together, all the c terms together.

We now have our answer:

Knowing how to take integrals when there are letters in the integrand will save us lots of time in problems like this next one.

Find  if

(a) a = 1, b = 2, c = 3

(b) a = 2, b = -1, c = 5

(c) a = -1, b = 1, c = 1

Since we already know that

we can plug each set of values for a, b, c into . We already found the integral; what's left is algebra.

(a) We substitute the values a = 1, b = 2, c = 3 to get

(b) We plug in a = 2, b = -1, c = 5 to get

(c) We plug in a = -1, b = 1, c = 1 to get