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Cogito

Calculus · Chapter 7 · Lesson 2

Practice Worksheet

Antiderivatives

Differentiation, run backwards.

12 problems · about 21 minutes · FUN-6.A

NameDate

Figure — use these to answer the problems

Change c and the curve slides up or down while its slope at every x stays identical. That is the constant of integration.
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y = 1x² + 0x + 0
a1
c, the constant0
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. Antiderivative of 8x is kx². What is k?

    Answer

  2. How can you always check an antiderivative?

    • Differentiate it and see if the original returns.
    • Compare the two graphs by eye.

  3. Antiderivative of x² is x³ ÷ k. What is k?

    Answer

  4. Antiderivative of x⁴ is x⁵ ÷ k. What is k?

    Answer

  5. Antiderivative of 6x² is kx³. What is k?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. Antiderivative of 5 is kx. What is k?

    Answer

  2. Antiderivative of 10x is kx². What is k?

    Answer

  3. For which exponent does the reversed Power Rule fail?

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Put the antidifferentiation steps in order.

    Write 1 to 4 in the boxes to put these in order.

    • Add one to the exponent.
    • Divide by the new exponent.
    • Add the constant of integration.
    • Differentiate the result to check.

  2. Antiderivative of 12x³ is kx⁴. What is k?

    Answer

  3. The Reverse

    The antiderivative of 2x is x² + C. What is the antiderivative of 4x, written as kx² + C? Give k.

    Answer

  4. The Constant

    Why does every antiderivative carry a + C? Enter 1 if constants vanish when differentiated, 2 if it is a notation habit.

    Answer