MAT137 Calculus! Lecture 31

Transcription

1 MAT137 Calculus! Lecture 31 Today: Next: Integration Methods: Integration Methods: Trig. Functions (v ) Rational Functions Trig. Substitution (v ) (v )

2 Integration by Parts Example 3 Evaluate x 2 sin x dx

3 Integration by Parts Example 3 Evaluate x 2 sin x dx Question How many Integrations by Parts do you think would be required to compute the integral for a positive integer n? x n sin x dx

4 Integration by Parts for Definite Integrals True or False? 3 0 xex, dx = xe x 3 0 ex dx

5 Integration by Parts for Definite Integrals If we combine the formula for Integration by Parts with the FTC, we can easily evaluate definite integrals via Integration by Parts. Theorem a b u dv = uv b a a b v du Homework Evaluate x 2 ln x dx. 1 2

6 Integration by Parts Example 1 Evaluate sec 3 x dx.

7 True or False cos 8 x dx = 1 9 cos9 x sin x + C

8 Method To compute sin n x cos m x dx: if, then try u = sin x; if, then try u = cos x. If both m and n are odd, you can try either substitution.

9 Method To compute sin n x cos m x dx: if m is odd, then try u = sin x; if, then try u = cos x. If both m and n are odd, you can try either substitution. Important Identity sin 2 x + cos 2 x = 1

10 Method To compute sin n x cos m x dx: if m is odd, then try u = sin x; if n is odd, then try u = cos x. If both m and n are odd, you can try either substitution. Important Identity sin 2 x + cos 2 x = 1

11 Exercise Describe a method to compute sec n x tan m x dx. When 1 m even. 2 n odd. Useful Identity tan 2 x + 1 = sec 2 x

12 Exercise Describe a method to compute sec n x tan m x dx. When 1 m even. 2 n odd. Useful Identity Method tan 2 x + 1 = sec 2 x To compute sec n x tan m x dx: if m is odd, then try ;

13 Exercise Describe a method to compute sec n x tan m x dx. When 1 m even. 2 n odd. Useful Identity Method tan 2 x + 1 = sec 2 x To compute sec n x tan m x dx: if m is odd, then try ; save a factor of sec x tan x and use tan 2 x = sec 2 x 1 to express the remaining factors in terms of sec x.

14 Exercise Describe a method to compute sec n x tan m x dx. When 1 m even. 2 n odd. Useful Identity Method tan 2 x + 1 = sec 2 x To compute sec n x tan m x dx: if m is odd, then try ; save a factor of sec x tan x and use tan 2 x = sec 2 x 1 to express the remaining factors in terms of sec x. if n is even, then try.

15 Exercise Describe a method to compute sec n x tan m x dx. When 1 m even. 2 n odd. Useful Identity Method tan 2 x + 1 = sec 2 x To compute sec n x tan m x dx: if m is odd, then try ; save a factor of sec x tan x and use tan 2 x = sec 2 x 1 to express the remaining factors in terms of sec x. if n is even, then try. save a factor of sec 2 x and use sec 2 x = tan 2 x + 1 to express the remaining factors in terms of tan x.

16 Exercise Evaluate sec x tan 2 x dx.

17 Exercise Evaluate sec x tan 2 x dx. Question How would you integrate sec n x tan m x dx if m is odd and n is even?

18 Integrals Featuring x 2 a 2, a 2 x 2, x 2 + a 2 Example 2 Find the area of the half disk of radius a.

19 Integrals Featuring x 2 a 2, a 2 x 2, x 2 + a 2 Does computing the definite integral Explain x 2 dx make sense?

20 Integrals Featuring x 2 a 2, a 2 x 2, x 2 + a 2 Evaluate 4x dx.

21 Integrals Featuring x 2 a 2, a 2 x 2, x 2 + a 2 Substitution x = a sec θ Domain 1 Let x = a sec θ where θ [0, π 2 ) ( π 2, π]. Evaluate x 2 a 2. 2 Let x = a sec θ where θ [0, π 2 ) [π, 3π 2 ). Evaluate x 2 a 2. Which domain makes your computations easier? Why?