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Cogito

AP Calculus BC · Chapter 6 · Lesson 1

Practice Worksheet

Integration by Parts

The Product Rule, run backwards.

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

NameDate

Figure — use these to answer the problems

Products of exponentials with polynomials are what this technique exists for.
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y = 1 · 2^x + 0
a1
b2
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. Integrating x⁵ times eˣ. How many applications of the formula?

    Answer

  2. How should u be chosen?

    • The factor that simplifies when differentiated.
    • Whichever factor is written first.

  3. Integrating x² times eˣ. How many applications?

    Answer

  4. Integrating x times sin x. Which should be u? 1 x, 2 sin x.

    Answer

  5. Integrating ln x. Which should be u? 1 ln x, 2 dx.

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. If u = x then du = k dx. What is k?

    Answer

  2. If dv = eˣ dx then v = k times eˣ. What is k?

    Answer

  3. Integrating x⁴ times eˣ. How many applications?

    Answer

Stretch Yourself

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

4 problems

  1. Put the integration by parts process in order.

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

    • Choose u as the factor that simplifies when differentiated.
    • Set dv to the rest and find v and du.
    • Apply the formula.
    • Evaluate the new integral, repeating if needed.

  2. Integrating x times cos x. Which should be u? 1 x, 2 cos x.

    Answer

  3. The Choice

    Integrating x times eˣ. Which should be u? 1 x, 2 eˣ.

    Answer

  4. The Repeats

    Integrating x³ times eˣ. How many applications of the formula are needed?

    Answer