Find the simplest antiderivative of the function. Pay attention to which variable is being used.
f(x) = 10x9
Answer
F(x) = x10
Example 2
Find the simplest antiderivative of the function. Pay attention to which variable is being used.
f(x) = x3
Answer
Example 3
Find the simplest antiderivative of the function. Pay attention to which variable is being used.
Answer
F(x) = ln |x| (we need the absolute value signs in this case to make sure F is defined everywhere that f is defined).
Example 4
Find the simplest antiderivative of the function. Pay attention to which variable is being used.
f(t) = -sin t
Answer
F(t) = cos t
Example 5
Find the simplest antiderivative of the function. Pay attention to which variable is being used.
f(x) = -cos x
Answer
F(x) = -sin x
Example 6
Find the simplest antiderivative of the function. Pay attention to which variable is being used.
f(t) = sec2t
Answer
F(t) = tan t
Example 7
Find the simplest antiderivative of the function. Pay attention to which variable is being used.
f(x) = e2x
Answer
Example 8
Find the simplest antiderivative of the function. Pay attention to which variable is being used.
f(x) = 5x ln 5
Answer
F(x) = 5x
Example 9
Find the simplest antiderivative of the function. Pay attention to which variable is being used.
Answer
F(x) = -x-1 or
Example 10
Find the simplest antiderivative of the function. Pay attention to which variable is being used.
f(x) = x2 + 3x – 5
Answer
Example 11
Evaluate the integral.
Answer
Example 12
Evaluate the integral.
Answer
Example 13
Evaluate the integral.
Answer
There are two ways to do this problem. The way that probably comes to mind first is to use the FTC:
The other way is to notice that x3 is an odd function. When we integrate any odd function from -a to a, we get 0. So when we integrate x3 from -1 to 1, we get 0.
Example 14
Evaluate the integral.
Answer
Example 15
Evaluate the integral.
Answer
Either 1 – e-1 or should be an acceptable answer. We like the one with the fraction better than the one with the negative exponent.
should be an acceptable answer. We like the one with the fraction better than the one with the negative exponent.
