2. Packet #25 Substitution and Integration by Parts (Again)
Math 180
Packet #25
Substitution and Integration by Parts (Again)

3. Substitution
When using the substitution method with definite integrals, change your limits to be in terms of 𝑢. Ex 1. Evaluate: −1 1 3 𝑥 2 𝑥 3 +1 𝑑𝑥

4. Ex 2. Evaluate: 𝜋/4 𝜋/2 cot 𝑥 csc 2 𝑥 𝑑𝑥 Ex 3
Ex 2. Evaluate: 𝜋/4 𝜋/2 cot 𝑥 csc 2 𝑥 𝑑𝑥 Ex 3. Evaluate: −𝜋/4 𝜋/4 tan 𝑥 𝑑𝑥

5. Integration by Parts
Ex 𝑥 𝑒 𝑥 𝑑𝑥 Ex 5. 𝜋 4 𝜋 2 cos 𝑥 sin 𝑥 ln sin 𝑥 𝑑𝑥

6. Recall that 𝑓(𝑥) is even if 𝑓 −𝑥 =𝑓(𝑥), and 𝑓(𝑥) is odd if 𝑓 −𝑥 =−𝑓(𝑥)
Recall that 𝑓(𝑥) is even if 𝑓 −𝑥 =𝑓(𝑥), and 𝑓(𝑥) is odd if 𝑓 −𝑥 =−𝑓(𝑥). If 𝑓 is even, then −𝑎 𝑎 𝑓(𝑥) 𝑑𝑥=2 0 𝑎 𝑓(𝑥) 𝑑𝑥. If 𝑓 is odd, then −𝑎 𝑎 𝑓(𝑥) 𝑑𝑥=0.

7. Recall that 𝑓(𝑥) is even if 𝑓 −𝑥 =𝑓(𝑥), and 𝑓(𝑥) is odd if 𝑓 −𝑥 =−𝑓(𝑥)
Recall that 𝑓(𝑥) is even if 𝑓 −𝑥 =𝑓(𝑥), and 𝑓(𝑥) is odd if 𝑓 −𝑥 =−𝑓(𝑥). If 𝑓 is even, then −𝑎 𝑎 𝑓(𝑥) 𝑑𝑥=2 0 𝑎 𝑓(𝑥) 𝑑𝑥. If 𝑓 is odd, then −𝑎 𝑎 𝑓(𝑥) 𝑑𝑥=0.

8. Recall that 𝑓(𝑥) is even if 𝑓 −𝑥 =𝑓(𝑥), and 𝑓(𝑥) is odd if 𝑓 −𝑥 =−𝑓(𝑥)
Recall that 𝑓(𝑥) is even if 𝑓 −𝑥 =𝑓(𝑥), and 𝑓(𝑥) is odd if 𝑓 −𝑥 =−𝑓(𝑥). If 𝑓 is even, then −𝑎 𝑎 𝑓(𝑥) 𝑑𝑥=2 0 𝑎 𝑓(𝑥) 𝑑𝑥. If 𝑓 is odd, then −𝑎 𝑎 𝑓(𝑥) 𝑑𝑥=0.

9. Ex 6. Evaluate: −2 2 ( 𝑥 4 −4 𝑥 2 +6) 𝑑𝑥 Ex 7. Evaluate: −𝜋 𝜋 sin 𝑥 𝑑𝑥