Cogito
AP Calculus BC · Chapter 6 · Lesson 3
Choosing an Integration Technique
Recognising which tool the integral wants.
12 problems · about 21 minutes · FUN-6.D
Figure — use these to answer the problems
- Point(0, 0)
- Second point(2, 4)
- Slope between them2
Warm Up
Straightforward practice. Get the method working first.
5 problemsx times sin x. Which technique? 1 substitution, 2 parts.
AnswerWhat signals substitution?
- A function and its derivative are both present.
- Any product of two functions.
x times eˣ. Which technique? 1 substitution, 2 parts.
Answerx times e^(x²). Which technique? 1 substitution, 2 parts.
Answer1 over ((x−1)(x−2)). Which technique? 1 partial fractions, 2 parts.
Answer
Build It Up
The same ideas with more to keep track of.
3 problemsln x on its own. Which technique? 1 substitution, 2 parts.
AnswerCan every integration answer be checked by differentiating? 1 yes, 0 no.
AnswerWhich is cheapest to try first? 1 substitution, 2 partial fractions.
Answer
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each integral by the technique it wants.
Write each item under the heading it belongs to: x times e to the x squared · 2x times (x squared plus 1) cubed · x times sin x · x times e to the x
Substitution
Integration by parts
2x times (x² + 1)³. Which technique? 1 substitution, 2 parts.
AnswerThe Signal
A function and its derivative are both present. Which technique? 1 substitution, 2 parts.
AnswerThe Rational
A rational function with a factorable denominator. Which technique? 1 partial fractions, 2 parts.
Answer