TABLE OF CONTENTS
Find where a is a constant greater than zero.
We need an antiderivative of xa. 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.
(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