Integration by parts

1. Integration by parts How to integrate some special products of two functions 纪光 - 北京景山学校

2. Problem • Not all functions can be integrated by using simple derivative formulas backwards … • Some functions look like simple products but cannot be integrated directly. Ex. f(x) = x .sin x but … u’.v’≠ (u.v)’ So what ?!? 纪光 - 北京景山学校

3. if g(x) = f[u(x)] and u and f have derivatives, then g’(x) = f’[u(x)] . u’(x) hence Reminder 1 Derivative of composite functions 纪光 - 北京景山学校

4. Examples of integrals products of composite functions 纪光 - 北京景山学校

5. Formulas of integrals of products made of composite functions 纪光 - 北京景山学校

6. If u and v have derivatives u’ and v’, • then • (u.v)’ = u’.v + u.v’ • u.v’ = (u.v)’ - u’.v • hence by integration of both sides Reminder 2 Derivative of the Product of 2 functions 纪光 - 北京景山学校

7. Integration by parts u = 1st part (to derive) v’ = 2nd part (to integrate) Example 1 u =x v’ = sin x u’ =(x)’ = 1 sin x = (- cos x)’ v = - cos x u’v = 1.(- cos x) 纪光 - 北京景山学校

8. Integration by parts u = xu’ = 1 v’ = sin x v = - cos x Example 1 纪光 - 北京景山学校

9. Integration by parts 1st part (to derive) 2nd part (to integrate) Example 2 u =x2 v’ = cos x u’ =(x2)’ = 2x cos x = (sin x)’ v =sinx u’v = 2x.sin x 纪光 - 北京景山学校

10. Integration by parts u = x2u’ = 2x v’ = cos x v = sin x Example 2 纪光 - 北京景山学校

11. Integration by parts 1st part (to derive) 2nd part (to integrate) Example 3 v’ =x u =ln x u’ =(ln x)’ = x = =>v = . u’v = 纪光 - 北京景山学校

12. Integration by parts u = ln xu’ = v’ = x v = Example 3 纪光 - 北京景山学校

13. Integration by parts 1st part (to derive) 2nd part (to integrate) Example 4 u =ln x v’ =1 u’ =(ln x)’ = 1 = (x)’=> v = u’v = 1 纪光 - 北京景山学校

14. Integration by parts u = ln xu’ = v’ = 1 v = x Example 4 纪光 - 北京景山学校

15. Integration by parts 1st part (to derive) 2nd part (to integrate) Example 5 u =ln x v’ =ln x u’ =(ln x)’ = ln x = (x ln x – x)’ v =x ln x – x u’v = ln x - 1 纪光 - 北京景山学校

16. Integration by parts u = ln xu’ = v’ = ln xv = x.ln x - x Example 5 纪光 - 北京景山学校

17. Integration by parts Use the IBP formula to calculate : Problem I 纪光 - 北京景山学校

18. Integration by parts Use the IBP formula to calculate : Problem II 纪光 - 北京景山学校

19. Integration by parts Let , for any Integer n ≥ 0 : Use the IBP formula to prove that : Problem III Then find I0, I1, I2 纪光 - 北京景山学校

20. Integration by parts Let , for any Integer n ≥ 0 : [Wallis Integral] Use the IBP formula to prove that : Calculate I0 and I1 Find a short formula for I2n Find a short formula for I2n+1 Problem IV 纪光 - 北京景山学校

21. Use twice the IBP formula to calculate Integration by parts Problem V 纪光 - 北京景山学校

22. Use twice the IBP formula to calculate Integration by parts Problem VI 纪光 - 北京景山学校

23. Integration by parts Problem VII Calculate I + J Use IBP to calculate I – J Find I and J 纪光 - 北京景山学校

24. Integration by parts Calculate I1 Find a relationship between Inand In-1 (n ≥ 1) Show that Show that Show that Problem VIII 纪光 - 北京景山学校