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Example 1 ln x x dx Example 1 ln x x dx We make the - - PowerPoint PPT Presentation

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Example 1 ln x x dx Example 1 ln x x dx We make the substitution: Example 1 ln x x dx We make the substitution: u = ln x Example 1 ln x x dx We make the substitution: du dx = 1 u = ln x , x Example 1 ln x x dx We

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SLIDE 1
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Example 1

ln x x dx

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Example 1

ln x x dx We make the substitution:

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Example 1

ln x x dx We make the substitution: u = ln x

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SLIDE 5

Example 1

ln x x dx We make the substitution: u = ln x, du dx = 1 x

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Example 1

ln x x dx We make the substitution: u = ln x, du dx = 1 x ln x x dx =

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SLIDE 7

Example 1

ln x x dx We make the substitution: u = ln x, du dx = 1 x ln x x dx =

  • u
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SLIDE 8

Example 1

ln x x dx We make the substitution: u = ln x, du dx = 1 x ln x x dx =

  • u.du

dx

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SLIDE 9

Example 1

ln x x dx We make the substitution: u = ln x, du dx = 1 x ln x x dx =

  • u.du

dx dx

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SLIDE 10

Example 1

ln x x dx We make the substitution: u = ln x, du dx = 1 x ln x x dx =

  • u.du

✚ ✚

dx✚

✚

dx

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SLIDE 11

Example 1

ln x x dx We make the substitution: u = ln x, du dx = 1 x ln x x dx =

  • u.du

✚ ✚

dx✚

✚

dx =

  • udu
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SLIDE 12

Example 1

ln x x dx We make the substitution: u = ln x, du dx = 1 x ln x x dx =

  • u.du

✚ ✚

dx✚

✚

dx =

  • udu = u2

2 + C

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SLIDE 13

Example 1

ln x x dx We make the substitution: u = ln x, du dx = 1 x ln x x dx =

  • u.du

✚ ✚

dx✚

✚

dx =

  • udu = u2

2 + C Now we substitute back:

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SLIDE 14

Example 1

ln x x dx We make the substitution: u = ln x, du dx = 1 x ln x x dx =

  • u.du

✚ ✚

dx✚

✚

dx =

  • udu = u2

2 + C Now we substitute back: ln x x dx =

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SLIDE 15

Example 1

ln x x dx We make the substitution: u = ln x, du dx = 1 x ln x x dx =

  • u.du

✚ ✚

dx✚

✚

dx =

  • udu = u2

2 + C Now we substitute back: ln x x dx = (ln x)2 2 + C

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SLIDE 16

Example 1

ln x x dx We make the substitution: u = ln x, du dx = 1 x ln x x dx =

  • u.du

✚ ✚

dx✚

✚

dx =

  • udu = u2

2 + C Now we substitute back: ln x x dx = (ln x)2 2 + C

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SLIDE 17

Example 2

  • ex

1 + e2x dx

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SLIDE 18

Example 2

  • ex

1 + e2x dx We can write this integral as:

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Example 2

  • ex

1 + e2x dx We can write this integral as:

  • ex

1 + e2x dx =

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SLIDE 20

Example 2

  • ex

1 + e2x dx We can write this integral as:

  • ex

1 + e2x dx =

  • ex

1 + (ex)2

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SLIDE 21

Example 2

  • ex

1 + e2x dx We can write this integral as:

  • ex

1 + e2x dx =

  • ex

1 + (ex)2 So, we make the substitution:

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Example 2

  • ex

1 + e2x dx We can write this integral as:

  • ex

1 + e2x dx =

  • ex

1 + (ex)2 So, we make the substitution: u = ex,

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SLIDE 23

Example 2

  • ex

1 + e2x dx We can write this integral as:

  • ex

1 + e2x dx =

  • ex

1 + (ex)2 So, we make the substitution: u = ex, du dx = ex

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SLIDE 24

Example 2

  • ex

1 + e2x dx We can write this integral as:

  • ex

1 + e2x dx =

  • ex

1 + (ex)2 So, we make the substitution: u = ex, du dx = ex

  • ex

1 + (ex)2 =

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Example 2

  • ex

1 + e2x dx We can write this integral as:

  • ex

1 + e2x dx =

  • ex

1 + (ex)2 So, we make the substitution: u = ex, du dx = ex

  • ex

1 + (ex)2 =

  • 1

1 + u2 .

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SLIDE 26

Example 2

  • ex

1 + e2x dx We can write this integral as:

  • ex

1 + e2x dx =

  • ex

1 + (ex)2 So, we make the substitution: u = ex, du dx = ex

  • ex

1 + (ex)2 =

  • 1

1 + u2 .du dx

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SLIDE 27

Example 2

  • ex

1 + e2x dx We can write this integral as:

  • ex

1 + e2x dx =

  • ex

1 + (ex)2 So, we make the substitution: u = ex, du dx = ex

  • ex

1 + (ex)2 =

  • 1

1 + u2 .du dx dx

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SLIDE 28

Example 2

  • ex

1 + e2x dx We can write this integral as:

  • ex

1 + e2x dx =

  • ex

1 + (ex)2 So, we make the substitution: u = ex, du dx = ex

  • ex

1 + (ex)2 =

  • 1

1 +

  • ✒

e2x

u2 .du dx dx

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Example 2

  • ex

1 + e2x dx We can write this integral as:

  • ex

1 + e2x dx =

  • ex

1 + (ex)2 So, we make the substitution: u = ex, du dx = ex

  • ex

1 + (ex)2 =

  • 1

1 + u2 .

✁ ✁ ✁ ✕

ex du dx dx

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Example 2

  • ex

1 + e2x dx We can write this integral as:

  • ex

1 + e2x dx =

  • ex

1 + (ex)2 So, we make the substitution: u = ex, du dx = ex

  • ex

1 + (ex)2 =

  • 1

1 + u2 .du dx dx

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SLIDE 31

Example 2

  • ex

1 + e2x dx We can write this integral as:

  • ex

1 + e2x dx =

  • ex

1 + (ex)2 So, we make the substitution: u = ex, du dx = ex

  • ex

1 + (ex)2 =

  • 1

1 + u2 .du

✚ ✚

dx✚

✚

dx

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SLIDE 32

Example 2

  • ex

1 + e2x dx We can write this integral as:

  • ex

1 + e2x dx =

  • ex

1 + (ex)2 So, we make the substitution: u = ex, du dx = ex

  • ex

1 + (ex)2 =

  • 1

1 + u2 .du

✚ ✚

dx✚

✚

dx =

  • du

1 + u2

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SLIDE 33

Example 2

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Example 2

  • du

1 + u2 =

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SLIDE 35

Example 2

  • du

1 + u2 = arctan u + C

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SLIDE 36

Example 2

  • du

1 + u2 = arctan u + C Substituting back:

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SLIDE 37

Example 2

  • du

1 + u2 = arctan u + C Substituting back: u = ex

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SLIDE 38

Example 2

  • du

1 + u2 = arctan u + C Substituting back: u = ex

  • ex

1 + e2x dx =

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SLIDE 39

Example 2

  • du

1 + u2 = arctan u + C Substituting back: u = ex

  • ex

1 + e2x dx = arctan ex + C

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Example 2

  • du

1 + u2 = arctan u + C Substituting back: u = ex

  • ex

1 + e2x dx = arctan ex + C

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SLIDE 41