13.2 integration by substitution
DESCRIPTION
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. Yah! We are done. What have you studied in MAT 116? What is Calculus?. What is “calculus”? - PowerPoint PPT PresentationTRANSCRIPT
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?