13.2 integration by substitution. let w be the inside function write do cross-multiply back to the...
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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?