Subject: Calculus

Constant Factor Rule In Integration

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;

\int c \cdot f(x)dx = c \cdot \int f(x)dx

for indefinite integrals and;

\int_{a}^{b} c \cdot f(x)dx = c \cdot \int_{a}^{b} 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.

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.

\int c \cdot [f(x) + g(x)] dx = c \cdot \int f(x)dx + c \cdot \int g(x)dx

for indefinite integrals and;

\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

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

Example #1

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

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+C.

Now, in the second integrand, the constant is 2z thus,

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

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

Answer:

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

Example #2

Solve: \int [x+z]dy

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

and

\int zdy=z \int dy = zy+C

And applying further sum rule, we then have

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

Answer:

(x+z)y+C

Example #3

Evaluate \int e^ydx

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 #4

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

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

Example #5

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

R\cot(e^xx^2)

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