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Cogito

AP Calculus AB · Chapter 6 · Lesson 3

Practice Worksheet

Substitution

The Chain Rule, run backwards.

12 problems · about 22 minutes · FUN-6.D

NameDate

Figure — use these to answer the problems

Substitution changes the variable, not the area. The shaded region is the same before and after.
-8-8-6-6-4-4-2-222446688
y = 1x² + 0x + 0
  • Point(0, 0)
  • Second point(3, 9)
  • Slope between them3
a1
Lower limit0
Upper limit3
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. If u = x⁴ + 3, then du = kx³ dx. What is k?

    Answer

  2. What must be done with the limits in a definite integral?

    • Change them to u values, or convert back to x before evaluating.
    • Leave them unchanged and evaluate in u.

  3. If u = x³ + 2, then du = kx² dx. What is k?

    Answer

  4. If u = 5x + 1, then du = k dx. What is k?

    Answer

  5. Integral of u³ du gives u⁴ ÷ k. What is k?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. Integral of u⁵ du gives u⁶ ÷ k. What is k?

    Answer

  2. Can a missing constant factor be compensated for? 1 yes, 0 no.

    Answer

  3. Can a missing variable factor be compensated for? 1 yes, 0 no.

    Answer

Stretch Yourself

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

4 problems

  1. Put the substitution process in order.

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

    • Choose u as the inside of the composition.
    • Differentiate to find du.
    • Rewrite the whole integral in terms of u.
    • Integrate, then convert back or change the limits.

  2. If u = x² − 7, then du = kx dx. What is k?

    Answer

  3. The Choice

    For the integral of 2x(x² + 1)³, what should u be? Enter 1 for x² + 1, 2 for 2x.

    Answer

  4. The Derivative

    If u = x² + 1, then du = kx dx. What is k?

    Answer