Calculus (I)
WEN-CHING LIEN
Department of Mathematics National Cheng Kung University
2008
WEN-CHINGLIEN Calculus (I)
8.2 Partial Fractions
f(x) = P(x)
Q(x), P,Q :polynomials
Step(1):Long Division before Integration.
Step(2):Now the rational function is proper.
Apply ”the partial fraction decomposition”.
8.2 Partial Fractions
f(x) = P(x)
Q(x), P,Q :polynomials
Step(1):Long Division before Integration.
Step(2):Now the rational function is proper.
Apply ”the partial fraction decomposition”.
WEN-CHINGLIEN Calculus (I)
8.2 Partial Fractions
f(x) = P(x)
Q(x), P,Q :polynomials
Step(1):Long Division before Integration.
Step(2):Now the rational function is proper.
Apply ”the partial fraction decomposition”.
8.2 Partial Fractions
f(x) = P(x)
Q(x), P,Q :polynomials
Step(1):Long Division before Integration.
Step(2):Now the rational function is proper.
Apply ”the partial fraction decomposition”.
WEN-CHINGLIEN Calculus (I)
Basic Principle: f (x ) = P (x ) Q(x) , (1)
If Q(x) contains a factor (ax + b)
n,
then the partial fraction decomposition contains
A
1ax + b + A
2(ax + b)
2+ . . .
. . . + A
n(ax + b)
n,
where A
1, A
2, . . . , A
nare constants.
Basic Principle: f (x ) = P (x ) Q(x) , (1)
If Q(x) contains a factor (ax + b)
n,
then the partial fraction decomposition contains
A
1ax + b + A
2(ax + b)
2+ . . .
. . . + A
n(ax + b)
n, where A
1, A
2, . . . , A
nare constants.
WEN-CHINGLIEN Calculus (I)
Basic Principle: f (x ) = P (x ) Q(x) , (1)
If Q(x) contains a factor (ax + b)
n,
then the partial fraction decomposition contains A
1ax + b + A
2(ax + b)
2+ . . .
. . . + A
n(ax + b)
n,
where A
1, A
2, . . . , A
nare constants.
Basic Principle: f (x ) = P (x ) Q(x) , (1)
If Q(x) contains a factor (ax + b)
n,
then the partial fraction decomposition contains A
1ax + b + A
2(ax + b)
2+ . . .
. . . + A
n(ax + b)
n, where A
1, A
2, . . . , A
nare constants.
WEN-CHINGLIEN Calculus (I)
Basic Principle: f (x ) = P (x ) Q(x) , (1)
If Q(x) contains a factor (ax + b)
n,
then the partial fraction decomposition contains A
1ax + b + A
2(ax + b)
2+ . . .
. . . + A
n(ax + b)
n,
where A
1, A
2, . . . , A
nare constants.
Basic Principle: f (x ) = P (x ) Q(x) , (1)
If Q(x) contains a factor (ax + b)
n,
then the partial fraction decomposition contains A
1ax + b + A
2(ax + b)
2+ . . .
. . . + A
n(ax + b)
n, where A
1, A
2, . . . , A
nare constants.
WEN-CHINGLIEN Calculus (I)
Example:
Z 1
(x +1)(x −3)dx
Examples:
1
Z x3+1 x2+3dx
2
Z 1
x2−9dx
3
Z 1
x2(x +1)dx
4
Z 1
(x2+x+1)2dx
WEN-CHINGLIEN Calculus (I)
Examples:
1
Z x3+1 x2+3dx
2
Z 1
x2−9dx
3
Z 1
x2(x +1)dx
4
Z 1
(x2+x+1)2dx
Examples:
1
Z x3+1 x2+3dx
2
Z 1
x2−9dx
3
Z 1
x2(x +1)dx
4
Z 1
(x2+x+1)2dx
WEN-CHINGLIEN Calculus (I)
Examples:
1
Z x3+1 x2+3dx
2
Z 1
x2−9dx
3
Z 1
x2(x +1)dx
4
Z 1
(x2+x+1)2dx
Examples:
1
Z x3+1 x2+3dx
2
Z 1
x2−9dx
3
Z 1
x2(x +1)dx
4
Z 1
(x2+x+1)2dx
WEN-CHINGLIEN Calculus (I)
(2)
If Q(x) contains an irreducible quadratic factor
(ax2+bx +c)n, then the partial fraction decomposition contains terms of the form:
B1x+C1
ax2+bx +c + B2x +C2
(ax2+bx+c)2+ . . .
. . .+ Bnx +Cn
(ax2+bx+c)n , (i.e. b2−4ac <0)
(2)
If Q(x) contains an irreducible quadratic factor
(ax2+bx +c)n,then the partial fraction decomposition contains terms of the form:
B1x+C1
ax2+bx +c + B2x +C2
(ax2+bx+c)2+ . . .
. . .+ Bnx +Cn
(ax2+bx+c)n , (i.e. b2−4ac <0)
WEN-CHINGLIEN Calculus (I)
(2)
If Q(x) contains an irreducible quadratic factor
(ax2+bx +c)n, then the partial fraction decomposition contains terms of the form:
B1x+C1
ax2+bx +c + B2x +C2
(ax2+bx+c)2+ . . .
. . .+ Bnx +Cn
(ax2+bx+c)n , (i.e. b2−4ac <0)
(2)
If Q(x) contains an irreducible quadratic factor
(ax2+bx +c)n, then the partial fraction decomposition contains terms of the form:
B1x+C1
ax2+bx +c + B2x +C2
(ax2+bx+c)2+ . . .
. . .+ Bnx +Cn
(ax2+bx+c)n , (i.e. b2−4ac <0)
WEN-CHINGLIEN Calculus (I)
(2)
If Q(x) contains an irreducible quadratic factor
(ax2+bx +c)n, then the partial fraction decomposition contains terms of the form:
B1x+C1
ax2+bx +c + B2x +C2
(ax2+bx+c)2+ . . .
. . .+ Bnx +Cn
(ax2+bx+c)n , (i.e. b2−4ac <0)
Examples:
1
Z 1
(x2+x+1)2dx
2 Computer the square in the denominator to evaluateZ 1 x2+2x +5dx
3
Z 1
(x2+9)2dx
4
Z 2x2+2x −1 x3(x −3) dx
WEN-CHINGLIEN Calculus (I)
Examples:
1
Z 1
(x2+x+1)2dx
2 Computer the square in the denominator to evaluateZ 1 x2+2x +5dx
3
Z 1
(x2+9)2dx
4
Z 2x2+2x −1 x3(x 3) dx
Examples:
1
Z 1
(x2+x+1)2dx
2 Computer the square in the denominator to evaluateZ 1 x2+2x +5dx
3
Z 1
(x2+9)2dx
4
Z 2x2+2x −1 x3(x −3) dx
WEN-CHINGLIEN Calculus (I)
Examples:
1
Z 1
(x2+x+1)2dx
2 Computer the square in the denominator to evaluateZ 1 x2+2x +5dx
3
Z 1
(x2+9)2dx
4
Z 2x2+2x −1 x3(x 3) dx
Examples:
1
Z 1
(x2+x+1)2dx
2 Computer the square in the denominator to evaluateZ 1 x2+2x +5dx
3
Z 1
(x2+9)2dx
4
Z 2x2+2x −1 x3(x −3) dx
WEN-CHINGLIEN Calculus (I)
Example:
I =Z 1
x4+1dx
Thank you.
WEN-CHINGLIEN Calculus (I)