Subject: Calculus

# Constant Factor Rule In Integration

## Calculus.ConstantFactorRuleInIntegration History

July 27, 2011 by matthew_suan -
Changed lines 93-94 from:
Calculate: {$\int d[R\cot{(e^xx^2)]$}
to:
Calculate: {$\int d[R\cot{(e^xx^2)}]$}
Changed line 98 from:
{$R\cot{(e^xx^2)$}
to:
{$R\cot{(e^xx^2)}$}
July 27, 2011 by matthew_suan -
Changed lines 48-49 from:
Evaluate {$\int [x+z]dy$}
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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{$R\cot{(e^xx^2)$}
July 27, 2011 by matthew_suan -
Changed line 64 from:
{$\int [x+z]dy=xy+zy+C$
to:
{$\int [x+z]dy=xy+zy+C$}
July 27, 2011 by matthew_suan -
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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to:
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to:
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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,

{$\int e^ydx=xe^y+C$}.
July 03, 2011 by matthew_suan -
Changed line 24 from:
!!Example
to:
!!!Example #1
!!!Example #2
'''NEXT TOPIC''': [[Linearity of integration]]
February 09, 2011 by matthew_suan -
Changed line 47 from:
Evaluate {$\int [x+z]dy$
to:
Evaluate {$\int [x+z]dy$}
February 09, 2011 by matthew_suan -
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:
Find the derivative of {$4x^3+2x+1$}
to:
Evaluate {$\int [x+z]dy$
Changed lines 52-54 from:
''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

{$\int xdy=x \int dy = xy+C$}
Changed lines 59-68 from:
{$\int 2x=x^2$}

and

{$\int 1=x$}

Thus,
according to the sum rule,

{$\int [4x^3+2x+1]dx=x^4+x^2+x$}
to:
{$\int zdy=z \int dy = zy+C$}

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 by matthew_suan -
Changed lines 25-26 from:
Find the derivative of {$x^2+2x$}
to:
Evaluate {$\int [x^2y+2zx]dx$}
Changed lines 30-32 from:
''Solution:'' We know that

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

{$\int x^2ydx=y \int x^2dx=y \cdot \frac{1}{3}x^3$}.
Changed lines 35-42 from:
and

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

Thus,
according to the sum rule,

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

{$\int [x^2y+2zx]dx=\int x^2ydx+\int 2zxdx=y \cdot \frac{1}{3}x^3+2z \cdot \frac{1}{2}x^2+C$}
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 by matthew_suan -
Deleted lines 0-2:
{$\int [af(x) + bg(x)] dx$} = {$a\int f(x)dx + b\int g(x)dx$}
February 09, 2011 by matthew_suan -
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%{$\int [h(x)] dx$} = {$\int f(x)dx + \int g(x)dx$}
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%{$\int c \cdot [f(x) + g(x)] dx$} = {$c \cdot \int f(x)dx + c \cdot \int g(x)dx$}
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 matthew_suan -
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 by matthew_suan -
{$\int [af(x) + bg(x)] dx$} = {$a\int f(x)dx + b\int g(x)dx$}

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 {$x^2+2x$}

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''Solution:'' We know that

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

and

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

Thus, according to the sum rule,

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

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

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

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''Solution:'' We know that

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

and

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

and

{$\int 1=x$}

Thus, according to the sum rule,

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

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