24. Tabular Integration

7/28/2019 24. Tabular Integration

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MatoMato TipilaTipila National Monument, WyomingNational Monument, Wyoming


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A Shortcut: Tabular Integration

Tabular integration works for integrals of the form:

f x g x dxwhere: Differentiates to

zero in severalsteps.

Integratesrepeatedly.

Recall:


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3 sinx x dx

3x

23x

6x6

sin x

cosx

sin x

cosx

0

sin x3 cosx x 23 sinx x 6 cosx x 6sin x + C

& deriv.f x & integralsg x


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6x6

0


dxexx x23 2

xx 23

23 2 x

xe

2

2

2xe

4

2xe

8

2xe

16

2xe

Cxxxex

714648

232


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3x

23x

6x

6

0


dxxx 2sin3

x2sin

22cos x

4

2sin x

16

2sin x

8

2cos x

C

xxxxxxx

8

2sin3

4

2cos3

4

2sin3

2

2cos 23


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The chain rule allows us to differentiatedifferentiate a widewide

varietyvariety of functions;

However, we are able to find antiderivativesantiderivatives for onlya limited rangelimited range of functions;

We can sometimes use substitutionsubstitution to rewriterewrite

functionsfunctions in a form that we can integrate.


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Example 1:

5

2x dx Let 2u x du dx

5

u du61

6 u C

6

26

x C

The variable of integrationmust match the variable inthe expression.

Dont forget to substitute the value

for u back into the problem!


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Example 2:

21 2x x dx

One of the clues that we look for isif we can find a function and itsderivative in the integrand.

The derivative of is .21 x 2x dx

1

2 u du3

22

3u C

3

2 22

1

3

x C

2Let 1u x

2du x dx

Note that this only worked becauseof the 2x in the original.Many integrals can not be done by

substitution.


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Example 3:

4 1x dx Let 4 1u x 4du dx

1

4du dx

Solve for dx.

1

2 14

u du3

22 13 4

u C

3

216

u C

3

21 4 16

x C


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Example 4:

cos 7 5x dx7du dx

1

7du dx

1cos7

u du

1 sin7

u C

1

sin 7 57

x C

Let 7 5u x


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Example 5:

2 3

sinx x dx3

Let u x23du x dx

21

3du x dx

We solve forbecause we can find itin the integrand.

2x dx

1 sin3

u du

1 cos3

u C

31 cos3

x C


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Example 6:

4

sin cosx x dx

Let sinu xcosdu x dx

4

sin cosx x dx4

u du51

5u C

51 sin5

x C


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Example 7:

240

tan secx x dx

The technique is a little differentfor definite integrals.

Let tanu x

2secdu x dx

0 tan 0 0u

tan 14 4

u

1

0u du We can find

new limits,

and then wedont haveto substitute

back.

new limit

new limit

1

2

0

1

2u

1

2

We could have substituted back andused the original limits.


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Example 7:

240

tan secx x dx

Let tanu x

2secdu x dx4

0 u du

Wrong!The limits dont match!

42

0

1

tan2 x

2

21 1tan tan 02 4 2

2 21 11 0

2 2

u du21

2u

1

2

Using the original limits:

Leave the

limits out untilyou substituteback.

This isusuallymore work

than findingnew limits


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Example 8:

12 3

13 x 1x dx 3

Let 1u x 23du x dx

1 0u 1 2u

12

2

0u du

23

2

0

23

u Dont forget to use the new limits.

3

22

23

2

2 23

4 2

3


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Separable Differential Equations

A separable differential equation can be expressed as

the product of a function ofx and a function ofy.

dy g x h ydx

Example:

22dy

xydx

Multiply both sides by dx and divide

both sides by y2 to separate the

variables. (Assumey2 is never zero.)

22dy x dx

y

2 2y dy x dx

0h y


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Separable Differential Equations

A separable differential equation can be expressed as

the product of a function ofx and a function ofy.

dy g x h ydx

Example:

22dy

xydx

22dy x dx

y

2 2y dy x dx

2

2y dy x dx

1 2

1 2y C x C

21 x Cy

2

1yx C 2

1y x C

0h y

Combinedconstants of

integration


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Example 9:

22

2 1xdy

x y edx

2

2

12

1

xdy x e dx

y

Separable differential equation

2

2

12

1

xdy x e dx

y

2

u x

2du x dx

2

1

1

udy e du

y

1

1 2tanu

y C e C

21

1 2

tan xy C e C

21tan xy e C Combined constants of integration


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Example 9:

22

2 1xdy

x y edx

2

1tan xy e C

We now havey as an implicitfunction ofx.

We can findy as an explicit functionofx by taking the tangent of bothsides.

2

1tan tan tan xy e C

2

tan

x

y e C

Notice that we can not factor out the constant C, becausethe distributive property does not work with tangent.


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Until then, remember that most college professors donot allow calculators.

In another generation or so, we might be able to usethe calculator to find all integrals.

You must practice finding integrals by hand until you aregood at it!

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24. Tabular Integration