Cogito
Calculus · Chapter 7 · Lesson 2
Antiderivatives
Differentiation, run backwards.
12 problems · about 21 minutes · FUN-6.A
What this lesson teaches
The student finds antiderivatives of basic functions and explains the constant of integration.
- An antiderivative is a function whose derivative is the given one.
- Reverse the Power Rule: add one to the exponent and divide by it.
- Every antiderivative carries + C, because constants differentiate away.
Warm Up
Straightforward practice. Get the method working first.
5 problemsAntiderivative of 8x is kx². What is k?
Answer 4
Why 4.
How can you always check an antiderivative?
Answer Differentiate it and see if the original returns.
Why Differentiate the answer.
Antiderivative of x² is x³ ÷ k. What is k?
Answer 3
Why The new exponent.
Antiderivative of x⁴ is x⁵ ÷ k. What is k?
Answer 5
Why The new exponent.
Antiderivative of 6x² is kx³. What is k?
Answer 2
Why 6 ÷ 3.
Build It Up
The same ideas with more to keep track of.
3 problemsAntiderivative of 5 is kx. What is k?
Answer 5
Why Differentiating 5x gives 5.
Antiderivative of 10x is kx². What is k?
Answer 5
Why 10 ÷ 2.
For which exponent does the reversed Power Rule fail?
Answer -1
Why Adding one would give zero.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the antidifferentiation steps in order.
Answer 1. Add one to the exponent. 2. Divide by the new exponent. 3. Add the constant of integration. 4. Differentiate the result to check.
Why The check is always last.
Antiderivative of 12x³ is kx⁴. What is k?
Answer 3
Why 12 ÷ 4.
The Reverse: The antiderivative of 2x is x² + C. What is the antiderivative of 4x, written as kx² + C? Give k.
Answer 2
Why 2.
The Constant: Why does every antiderivative carry a + C? Enter 1 if constants vanish when differentiated, 2 if it is a notation habit.
Answer 1
Why Constants differentiate to zero, so they are invisible going forwards.