13.2 Integration by Substitution

• Let w be the inside function

• Write

• Do cross-multiply • Back to the given integral and do the

substitution• Take the integral• Express the final answer in term of x

)(' xwdx

dw

?)6()32( 2203 dxxx

26xdx

dw

dxxdw 26

dww

dxxx

20

2203 )6()32(

?5 5 dxe x

5dx

dw

dxdw 5

dwe

dxe

w

x

55

Cew

Ce x 5

?24

12

xdxx

xdx

dw2

xdxdw 2

dww

xdxx

1

24

12

Cw ln

Cx 24ln

32 xdx

dw

dxxdw )32( C

w

5

5

Cxx

5

)73( 52

?)32()73( 42 dxxxx

dww

dxxxx

4

42 )32()73(

?)6()32( 243 dxxx

dww

dxxx

4

243 )6()32(

Cw

5

526x

dx

dw

dxxdw 26C

x

5

)32( 53

?3 dxe x

dwe

dxe

w

x

3

1

3

dwew

3

1

Cew

3

1

3dx

dw

dxdw 3

dxdw

3

dxdw 3

1 Ce x

3

3

1

dww

dxx

x

2

1192

dww 1

2

1

Cw ln2

1

xdx

dw2

xdxdw 2

xdxdw

2

xdxdw2

1 Cx 9ln2

1 2

dxx

x 92

)100(001.

001. 01.

dwe

dxe

w

x

dwew 1.0

Cew 1.0

01.dx

dw

dxdw 01.

dxdw

01.

dxdw100Ce x 01.1.0

?001.0 01.0 dxe x

• Yah! We are done.• What have you studied in MAT 116?• What is Calculus?

• What is “calculus”?– Calculus is the study of how things change.

• What do you learn in a calculus class?– Functions, limits, derivatives, integrals, …

• Derivatives can used to study applications that involve the rate of change such as population growth, radioactive decay, optimization problems, …

• Integrals can be used to calculate the area, equation of a curve, business sales, …

• How do algebra and calculus differ?

Ex:Solving equations to find a particular value of a variable

Ex:How a change in a variable affects another variable?