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ENGI 3424 Tutorial – Integration by Parts...

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ENGI 3424 1.9

Tutorial – Integration by Parts

Page 1-56

Integration by Parts [Tutorial in Week 2 or 3]

Review: Let u  x  and v x  be functions of x. Then, by the product rule of differentiation, d du dv  uv   v  u  dx dx dx

Integrating with respect to x :

uv 



 uv dx   uv  dx

This leads to the formula for integration by parts:

 uv dx Example 1.9.1 Find

I 



2

x 3e  x dx .

 uv  

 uv dx

ENGI 3424

Tutorial – Integration by Parts

Example 1.9.2 Find

I 

[Repeated use of integration by parts]

 x cos x dx . 2

Page 1-57

ENGI 3424

Tutorial – Integration by Parts

Example 1.9.2 (continued) Shortcut (a tabular form for repeated integrations by parts):

I 

x

2

cos x dx :

Example 1.9.3 Find

I 

x e 4

x

dx .

Page 1-58

ENGI 3424

Tutorial – Integration by Parts

Example 1.9.4 Find

I 

e

(recursive use of integration by parts) ax

sinbx dx ,

(where a, b are constants).

Page 1-59

ENGI 3424 Example 1.9.4 (continued)

Tutorial – Integration by Parts

Page 1-60

ENGI 3424

Tutorial – Integration by Parts

Page 1-61

Example 1.9.5 Find I n  x  

x

n

ln x dx , where n is any number except –1.

The derivative of ln x is known, but not the antiderivative (unless integration by parts is used!). This is therefore a rare instance where a polynomial factor is not in the left [differentiation] column.

END OF THE APPENDIX TO CHAPTER 1

ENGI 3424

Tutorial – Integration by Parts [Space for additional notes]

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