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Title: Integration of quadratic polynomials
Description: This note contains worked examples of integrals containing quadratic polynomials
Description: This note contains worked examples of integrals containing quadratic polynomials
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[Last Name] 1
INTEGRATION
INTEGRALS CONTAINING QUADRATIC POLYNOMIALS (SPECIAL CASES)
CASE 1
π
Many Integrals in these general forms, β« (πππ+ππ+π) π π,
β«(
π
) π π, πππ ππ ππππππππππ ππ πππ πππππππ ππ ππππππππππ πππ ππππππ
...
π
Worked Example 1: Find β« (πππ+ππ+π) π π
Completing the square
9xΒ²+6x+5
π (ππ +
ππ
π (ππ +
ππ
π
π
)+π
)+π
π π
π
π ((π + π) β π) + π
π π
π (π + ) β π + π
π
π π
π (π + π) + π
π π
ππ (π + ) + π
π
π
π
(π (π + π)) + π
(ππ + π)π + π
Re-writing the Integral
[Last Name] 2
π
β« ((ππ+π)π+π) π π
Applying the substitution, U=3x+1
π π
π π
= π
Making βdxβ the subject
π π =
π π
π
π
= π π π
Substituting the value of βdxβ into the Integral and re-writing the Integral in terms of U
π
π
β« (ππ+π)
...
π π π
Pulling out Β½
π
π
β«(
π
βππβππ
) π π
Integrating by trigonometric substitution
Recall, β« (
π
βππ βππ
π
) π π = πππβπ (π) + π
Re-writing the Integral in the right pattern for the trigonometric substitution
π
π
β«(
π
βππ βππ
) π π
x= u
a=5
π
π
π
πππβπ (π) + π
But, u= 2x-4
β«(
π
π
βπ+πππβπππ
) π π = π πππβπ (
ππβπ
π
)+π
CASE 2
Integrals in the form β« (
π
π π
π¨πΏ+π©
π
(πππ +ππ+π)
) π π, πππππ n ππ π ππππππππ πππππππ πππ πππππ
(πππ + ππ + π) β π¨π + π©, πππ ππ ππππππ ππ πππππππππ ππ ππππ πππππππ πππππππππ
that are in standard form
...
[Last Name] 6
We are going to re-write 2x+3 as a constant multiple of 18x+6
2x+3=P(18x+6)+Q
2x+3=18Px+6P+Q
Equating corresponding coefficients
2=18Pβ¦β¦(I)
6P+Q=3β¦
...
π π π
π
π
π
π·ππππππ πππ π πππ πππππππππππ ππ ππππ π
π
π
β«(
π
) π π
ππ +π
Integrating by trigonometric substitution
π
π
π
Recall, β« (ππ+ππ) π π = π πππβπ (π) + π
Re-writing the Integral in the right pattern for the trigonometric substitution
π
π
π
β« (ππ+ππ) π π
[Last Name] 8
x= u
a=2
π π
π
[ πππβπ ( π)] + π
π π
But, U=3x+1
Further simplifying
π
ππ
πππβπ (
ππ+π
) + ππ
π
π
π
π
β« ( πππ+ππ+π
) π π = ππ πππβπ (
ππ+π
π
) + ππ
Therefore,
ππ+π
π
π
β« (πππ+ππ+π) π π = π ππ(πππ + ππ + π) + ππ πππβπ (
ππ+π
π
)+πͺ
NOTE: ππ πππ ππ ππππ ππππ ππππππππ ππππ π ππππππ ππππππππ πͺ
WORKED EXAMPLE 2: Find β« (
(π+π)
π
(ππ +ππ+π)
) π π
π
In this case, π π (ππ + ππ + π) = ππ + π
We would need to re-write x+1 as a constant multiple of 2x+4
x+1=P(2x+4)+Q
Further simplifying
x+1=2Px+4P+Q
Equating corresponding coefficients
2P=1β¦
...
(II)
Solving simultaneously
[Last Name] 9
P=Β½
Q= -1
x+1 = P(2x+4)+Q
x+1= Β½(2x+4)-1
Re-writing the Integral
β«(
π
π
( )(ππ+π)βπ
π
(ππ +ππ+π)
) π π
Splitting into two simpler Integrals
π
β«(
(π)(ππ+π)
π
(ππ +ππ+π)
) π π β β« (
π
π
(ππ +ππ+π)
) π π
Start with the first Integral on the left
π
β«(
(π)(ππ+π)
π
(ππ +ππ+π)
) π π
Pulling out Β½
π
π
β«(
(ππ+π)
π
(ππ +ππ+π)
) π π
Applying the substitution U=ππ + ππ + π
π = ππ + ππ + π
π π
π π
= ππ + π
Making βdxβ the subject
π π
π
π π = ππ+π = ππ+π π π
Substituting the value of βdxβ into the Integral and re-writing the Integral in terms of U
π
π
β«(
ππ+π
π
πΌπ
ππ+π
)
...
π βπ+π
π
β ππΌ
But, U= xΒ²+4x+5
π
β«(
(π)(ππ+π)
π
π
(ππ +ππ+π)
) π π = β π(ππ+ππ+π) + ππ
Now, solving the second integral
ββ« (
(π)
π
(ππ +ππ+π)
) π π
Completing the square
xΒ²+4x+5
(xΒ²+4x)+5
((x+2)Β²-4)+5
(x+2)Β²-4+5
(x+2)Β²+1
Re-writing the Integral
β«(
π
π
((π+π)π +π)
) π π
Applying the substitution U=x+2
U=x+2
π π
π π
=π
Making βdxβ the subject
π π = π π
[Last Name] 11
Substituting the value of βdxβ into the Integral and re-writing the Integral in terms of U
β«(
π
π
(ππ +π)
) π π
Integrating by trigonometric substitution
Recall,
β«(
π
π
π
(ππ +ππ )
π
π
) π π = πππ (ππ+ππ) + πππ πππβπ (π) + π
a=1
x=u
Substituting the values of βaβ and βxβ it becomes;
ββ« (
π
π+π
π
(ππ +ππ+π)
Therefore, β« (
π
) π π = β π(ππ+ππ+π) β π πππ(π + π) + πͺπ
π+π
π
(ππ +ππ+π)
π
π+π
π
) π π = β π(ππ+ππ+π) β π(ππ+ππ+π) β π πππ(π + π) + π
βπβπ
π
= π(ππ+ππ+π) β π πππ(π + π) + πͺ
NOTE: ππ πππ ππ ππππ ππππ ππππππππ ππππ π ππππππ ππππππππ πͺ
Title: Integration of quadratic polynomials
Description: This note contains worked examples of integrals containing quadratic polynomials
Description: This note contains worked examples of integrals containing quadratic polynomials