Subject: Calculus

# Constant Factor Rule In Integration

## Calculus.ConstantFactorRuleInIntegration History

Hide minor edits - Show changes to output

July 27, 2011
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Calculate: {$\int d[R\cot{(e^xx^2)]$}

to:

Calculate: {$\int d[R\cot{(e^xx^2)}]$}

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{$R\cot{(e^xx^2)$}

to:

{$R\cot{(e^xx^2)}$}

July 27, 2011
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to:

Solve: {$\int [x+z]dy$}

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!!!Example #4

Integrate: {$\int_{0}^{1} kx^2+3$}

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{$\int_{0}^{1} [kx^2+3]dx = \frac{k}{3}+3$}

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!!!Example #5

Calculate: {$\int d[R\cot{(e^xx^2)]$}

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{$R\cot{(e^xx^2)$}

!!!Example #4

Integrate: {$\int_{0}^{1} kx^2+3$}

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{$\int_{0}^{1} [kx^2+3]dx = \frac{k}{3}+3$}

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!!!Example #5

Calculate: {$\int d[R\cot{(e^xx^2)]$}

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{$R\cot{(e^xx^2)$}

July 27, 2011
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Changed lines 14-15 from:

Constant factor rule in integration can also be applied to sum rule in integration. This will be given by the formula below.

to:

[[Constant factor rule in integration]] can also be applied to [[sum rule in integration]]. This will be given by the formula below.

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!!!Example #3

Evaluate {$\int e^ydx$}

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''Solution:''

Take note that {$y$} is a constant since the variable of integration is {$dx$}. Thus, by virtue of the constant factor rule,

{$\int e^ydx=xe^y+C$}.

!!!Example #3

Evaluate {$\int e^ydx$}

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''Solution:''

Take note that {$y$} is a constant since the variable of integration is {$dx$}. Thus, by virtue of the constant factor rule,

{$\int e^ydx=xe^y+C$}.

July 03, 2011
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!!~~Example~~

to:

!!!Example #1

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!!!Example #2

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'''NEXT TOPIC''': [[Linearity of integration]]

February 09, 2011
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Changed line 33 from:

{$\int x^2ydx=y \int x^2dx=y \cdot \frac{1}{3}x^3$}.

to:

{$\int x^2ydx=y \int x^2dx=y \cdot \frac{1}{3}x^3+C$}.

Changed lines 37-38 from:

{$\int 2zxdx=2z \int xdx=2z \cdot \frac{1}{2}x^2$}.

to:

{$\int 2zxdx=2z \int xdx=2z \cdot \frac{1}{2}x^2+C$}.

Changed lines 47-48 from:

to:

Evaluate {$\int [x+z]dy$

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''Solution:'' ~~We~~ ~~know~~ ~~that~~

{$~~\int~~ ~~4x^3=x^4$}~~

to:

''Solution:''

The variable of integration is {$dx$}, thus the all the integrands are constant. Then, constant factor rule proceeds by

{$\int xdy=x \int dy = xy+C$}

The variable of integration is {$dx$}, thus the all the integrands are constant. Then, constant factor rule proceeds by

{$\int xdy=x \int dy = xy+C$}

Changed lines 59-68 from:

{$\int ~~2x~~=~~x^2$}~~

and

{$\int ~~1~~=~~x$}~~

Thus, ~~according~~ ~~to~~ ~~the~~ sum rule,

{$\int [~~4x^3~~+~~2x+1~~]~~dx~~=~~x^4~~+~~x^2~~+~~x~~$~~}~~

and

{$\int

Thus,

{$\int

to:

{$\int zdy=z \int dy = zy+C$}

And applying further sum rule, we then have

{$\int [x+z]dy=xy+zy+C$

And applying further sum rule, we then have

{$\int [x+z]dy=xy+zy+C$

Changed line 66 from:

{$x~~^4~~+~~x^2~~+~~x~~$}

to:

{$(x+z)y+C$}

February 09, 2011
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Changed lines 25-26 from:

to:

Evaluate {$\int [x^2y+2zx]dx$}

Changed lines 30-32 from:

''Solution:'' ~~We~~ ~~know~~ that

{$\int ~~x^2=\frac~~{~~1~~}~~{3}x^3$}~~

{$\int

to:

''Solution:''

Take note that in the first integrand, {$y$} is a constant since the variable of integration is {$dx$}. Thus, by virtue of the constant factor rule, the first integrand is

{$\int x^2ydx=y \int x^2dx=y \cdot \frac{1}{3}x^3$}.

Take note that in the first integrand, {$y$} is a constant since the variable of integration is {$dx$}. Thus, by virtue of the constant factor rule, the first integrand is

{$\int x^2ydx=y \int x^2dx=y \cdot \frac{1}{3}x^3$}.

Changed lines 35-42 from:

{$\int

Thus,

to:

Now, in the second integrand, the constant is {$2z$} thus,

{$\int 2zxdx=2z \int xdx=2z \cdot \frac{1}{2}x^2$}.

Applying further the sum rule, we now have

{$\int [x^2y+2zx]dx=\int x^2ydx+\int 2zxdx=y \cdot \frac{1}{3}x^3+2z \cdot \frac{1}{2}x^2+C$}

{$\int 2zxdx=2z \int xdx=2z \cdot \frac{1}{2}x^2$}.

Applying further the sum rule, we now have

{$\int [x^2y+2zx]dx=\int x^2ydx+\int 2zxdx=y \cdot \frac{1}{3}x^3+2z \cdot \frac{1}{2}x^2+C$}

Changed line 44 from:

{$\frac{1}{3}x^3+x^2$}

to:

{$y \cdot \frac{1}{3}x^3+2z \cdot \frac{1}{2}x^2+C$}

February 09, 2011
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Deleted lines 0-2:

February 09, 2011
by -

Changed lines 12-13 from:

%cframe width=400px%{$ \int c \cdot f(x)dx $} = {$c \cdot \int f(x)dx$}

to:

%cframe width=400px%{$ \int_{a}^{b} c \cdot f(x)dx $} = {$c \cdot \int_{a}^{b} f(x)dx$}

Changed lines 16-20 from:

!!~~Alternative~~ ~~expression~~

Some ~~books~~ ~~has~~ ~~different~~ ~~expression~~ ~~with~~ ~~regards~~ ~~to~~ ~~this~~ ~~rule~~ ~~but~~ ~~basically~~ ~~they~~ ~~are~~ ~~just~~ ~~the~~ ~~same.~~ ~~Suppose~~ ~~we~~ ~~have~~ ~~a~~ ~~function~~ ~~{$h$}~~ ~~which~~ ~~can~~ ~~be~~ ~~expressed as {$h~~=~~g+f~~$~~}.~~ ~~Then~~ ~~the~~ ~~sum~~ ~~rule~~ ~~maintains~~ ~~that;~~

%cframe ~~width=400px%{$~~ ~~\int~~ ~~[h(x)]~~ ~~dx~~ ~~$}~~ ~~=~~ ~~{$ ~~\int f(x)dx + \int g(x)dx $}

Some

%cframe

to:

!!Constant factor rule and sum rule

Constant factor rule in integration can also be applied to sum rule in integration. This will be given by the formula below.

%cframe width=350px%{$ \int c \cdot [f(x) + g(x)] dx $} = {$ c \cdot \int f(x)dx + c \cdot \int g(x)dx $}

Constant factor rule in integration can also be applied to sum rule in integration. This will be given by the formula below.

%cframe width=350px%{$ \int c \cdot [f(x) + g(x)] dx $} = {$ c \cdot \int f(x)dx + c \cdot \int g(x)dx $}

Changed lines 23-24 from:

%cframe width=400px%{$ \int_{a}^{b} [~~h~~(x)~~]~~ ~~dx~~ $} = {$ \int_{a}^{b} f(x)dx + \int_{a}^{b} g(x)dx $}

to:

%cframe width=400px%{$ \int_{a}^{b} c \cdot [f(x) + g(x)] dx $} = {$ c \cdot \int_{a}^{b} f(x)dx + c \cdot \int_{a}^{b} g(x)dx $}

Deleted lines 25-38:

!!Another alternative expression

Even though the rule explicitly says "sum" rule, it does literally mean that it is exclusive to integrals with addition operation only. Remember that subtraction can also be expressed as an addition and is also therefore governed by the sum rule. All assumptions above imply such that the sum rule is also expressed as;

%cframe width=400px%{$ \int [f(x) - g(x)] dx $} = {$ \int f(x)dx - \int g(x)dx $}

for indefinite integrals and;

%cframe width=400px%{$ \int_{a}^{b} [f(x) - g(x)] dx $} = {$ \int_{a}^{b} f(x)dx - \int_{a}^{b} g(x)dx $}

for definite integrals since {$f(x)-g(x)=f(x)+[-g(x)]$}.

!!!Remark:

Even though we express the sum as the sum of two functions {$g$} and {$f$} in our expressions above, sum rule does not limit itself to sum of two functions only. The rule extends to sums of infinite number of functions as long as they are well-behaved.

February 09, 2011
by -

Changed lines 8-9 from:

%cframe width=400px%{$ \int c \~~cdotf~~(x)dx $} = {$c \cdot \int f(x)dx$}

to:

%cframe width=400px%{$ \int c \cdot f(x)dx $} = {$c \cdot \int f(x)dx$}

Changed line 12 from:

%cframe width=400px%{$ \int c \~~cdotf~~(x)dx $} = {$c \cdot \int f(x)dx$}

to:

%cframe width=400px%{$ \int c \cdot f(x)dx $} = {$c \cdot \int f(x)dx$}

February 09, 2011
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Added lines 1-86:

{$ \int [af(x) + bg(x)] dx $} = {$ a\int f(x)dx + b\int g(x)dx $}

The analog to [[sum rule in differentiation]] is the [[sum rule in integration]] which we discussed previously and since we have [[constant factor rule in differentiation]], then the analog for this in anti-derivative is the constant factor rule in integration. Basically, constant factor rule is not that complex. Here, we'll talk about the fundamentals of constant factor rule with regards to integration and explore its possible role in the previous rules in integration.

Constant factor rule in integration states that a constant factor inside an integrand can be separated from the integrand and instead multiplied by the integral. This statement can be best expressed through a mathematical statement. Consider a well-defined function {$f$} and a constant {$c$} such that constant factor rule applies as;

%cframe width=400px%{$ \int c \cdotf(x)dx $} = {$c \cdot \int f(x)dx$}

for indefinite integrals and;

%cframe width=400px%{$ \int c \cdotf(x)dx $} = {$c \cdot \int f(x)dx$}

for definite integrals given that both functions are well-defined at the interval [a, b]. Notice that along with the sum rule, constant factor rule is also true to both definite integrals and indefinite integrals as given above.

!!Alternative expression

Some books has different expression with regards to this rule but basically they are just the same. Suppose we have a function {$h$} which can be expressed as {$h=g+f$}. Then the sum rule maintains that;

%cframe width=400px%{$ \int [h(x)] dx $} = {$ \int f(x)dx + \int g(x)dx $}

for indefinite integrals and;

%cframe width=400px%{$ \int_{a}^{b} [h(x)] dx $} = {$ \int_{a}^{b} f(x)dx + \int_{a}^{b} g(x)dx $}

for definite integrals given that both functions are well-defined at the interval [a, b].

!!Another alternative expression

Even though the rule explicitly says "sum" rule, it does literally mean that it is exclusive to integrals with addition operation only. Remember that subtraction can also be expressed as an addition and is also therefore governed by the sum rule. All assumptions above imply such that the sum rule is also expressed as;

%cframe width=400px%{$ \int [f(x) - g(x)] dx $} = {$ \int f(x)dx - \int g(x)dx $}

for indefinite integrals and;

%cframe width=400px%{$ \int_{a}^{b} [f(x) - g(x)] dx $} = {$ \int_{a}^{b} f(x)dx - \int_{a}^{b} g(x)dx $}

for definite integrals since {$f(x)-g(x)=f(x)+[-g(x)]$}.

!!!Remark:

Even though we express the sum as the sum of two functions {$g$} and {$f$} in our expressions above, sum rule does not limit itself to sum of two functions only. The rule extends to sums of infinite number of functions as long as they are well-behaved.

!!Example

Find the derivative of {$x^2+2x$}

(:toggle id=box1 show='Show Solution and Answer' init=hide button=1:)

>>id=box1 border='1px solid #999' padding=5px bgcolor=#edf<<

''Solution:'' We know that

{$\int x^2=\frac{1}{3}x^3$}

and

{$\int 2x=x^2$}.

Thus, according to the sum rule,

{$\int [x^2+2x]dx=\frac{1}{3}x^3+x^2$}

!!!!Answer:

{$\frac{1}{3}x^3+x^2$}

>><<

Find the derivative of {$4x^3+2x+1$}

(:toggle id=box2 show='Show Solution and Answer' init=hide button=1:)

>>id=box2 border='1px solid #999' padding=5px bgcolor=#edf<<

''Solution:'' We know that

{$\int 4x^3=x^4$}

and

{$\int 2x=x^2$}

and

{$\int 1=x$}

Thus, according to the sum rule,

{$\int [4x^3+2x+1]dx=x^4+x^2+x$}

!!!!Answer:

{$x^4+x^2+x$}

>><<

The analog to [[sum rule in differentiation]] is the [[sum rule in integration]] which we discussed previously and since we have [[constant factor rule in differentiation]], then the analog for this in anti-derivative is the constant factor rule in integration. Basically, constant factor rule is not that complex. Here, we'll talk about the fundamentals of constant factor rule with regards to integration and explore its possible role in the previous rules in integration.

Constant factor rule in integration states that a constant factor inside an integrand can be separated from the integrand and instead multiplied by the integral. This statement can be best expressed through a mathematical statement. Consider a well-defined function {$f$} and a constant {$c$} such that constant factor rule applies as;

%cframe width=400px%{$ \int c \cdotf(x)dx $} = {$c \cdot \int f(x)dx$}

for indefinite integrals and;

%cframe width=400px%{$ \int c \cdotf(x)dx $} = {$c \cdot \int f(x)dx$}

for definite integrals given that both functions are well-defined at the interval [a, b]. Notice that along with the sum rule, constant factor rule is also true to both definite integrals and indefinite integrals as given above.

!!Alternative expression

Some books has different expression with regards to this rule but basically they are just the same. Suppose we have a function {$h$} which can be expressed as {$h=g+f$}. Then the sum rule maintains that;

%cframe width=400px%{$ \int [h(x)] dx $} = {$ \int f(x)dx + \int g(x)dx $}

for indefinite integrals and;

%cframe width=400px%{$ \int_{a}^{b} [h(x)] dx $} = {$ \int_{a}^{b} f(x)dx + \int_{a}^{b} g(x)dx $}

for definite integrals given that both functions are well-defined at the interval [a, b].

!!Another alternative expression

Even though the rule explicitly says "sum" rule, it does literally mean that it is exclusive to integrals with addition operation only. Remember that subtraction can also be expressed as an addition and is also therefore governed by the sum rule. All assumptions above imply such that the sum rule is also expressed as;

%cframe width=400px%{$ \int [f(x) - g(x)] dx $} = {$ \int f(x)dx - \int g(x)dx $}

for indefinite integrals and;

%cframe width=400px%{$ \int_{a}^{b} [f(x) - g(x)] dx $} = {$ \int_{a}^{b} f(x)dx - \int_{a}^{b} g(x)dx $}

for definite integrals since {$f(x)-g(x)=f(x)+[-g(x)]$}.

!!!Remark:

Even though we express the sum as the sum of two functions {$g$} and {$f$} in our expressions above, sum rule does not limit itself to sum of two functions only. The rule extends to sums of infinite number of functions as long as they are well-behaved.

!!Example

Find the derivative of {$x^2+2x$}

(:toggle id=box1 show='Show Solution and Answer' init=hide button=1:)

>>id=box1 border='1px solid #999' padding=5px bgcolor=#edf<<

''Solution:'' We know that

{$\int x^2=\frac{1}{3}x^3$}

and

{$\int 2x=x^2$}.

Thus, according to the sum rule,

{$\int [x^2+2x]dx=\frac{1}{3}x^3+x^2$}

!!!!Answer:

{$\frac{1}{3}x^3+x^2$}

>><<

Find the derivative of {$4x^3+2x+1$}

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>>id=box2 border='1px solid #999' padding=5px bgcolor=#edf<<

''Solution:'' We know that

{$\int 4x^3=x^4$}

and

{$\int 2x=x^2$}

and

{$\int 1=x$}

Thus, according to the sum rule,

{$\int [4x^3+2x+1]dx=x^4+x^2+x$}

!!!!Answer:

{$x^4+x^2+x$}

>><<