Subject: Calculus

# Linearity Of Integration

Linearity of integration is a direct consequence of sum rule in integration and constant factor rule in integration. Moreover, linearity of integration is one of the fundamental rules of calculus that takes its form according to the idea that integrations are just sums. Now, consider we have two functions f and g such that c and k are constants, then, the linearity of integration states that

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

for indefinite integrals and;

\int_{a}^{b} [c \cdot f(x) + k \cdot g(x)]dx=c \cdot \int_{a}^{b} f(x)dx + k \cdot \int_{a}^{b} g(x)dx

for definite integrals provided that both functions behave properly at the interval [a, b].

## Proof

Consider f and g as functions and c and k as constants such that

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

Applying sum rule in integration, we get

\int c \cdot f(x)dx + \int k \cdot g(x)dx

Getting constant factor rule in integration into the mix, the equation above becomes

c \cdot \int f(x)dx + k \cdot \int g(x)dx

Equating this to our original equation, we have

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

which is just the expression of the linearity of integration rule outlined in the first part above.

### Example #1

What is the fundamental rule of calculus that considers the idea that integration are just sums?

### Example #2

Linearity of integration is a direct consequence of the two fundamental rule of calculus. Name them.

1. sum rule in integration
2. constant factor rule in integration

### Example #3

Express the theorem of linearity of integration for definite integrals.

\int_{a}^{b} [c \cdot f(x) + k \cdot g(x)]dx=c \cdot \int_{a}^{b} f(x)dx + k \cdot \int_{a}^{b} g(x)dx

### Example #4

Express the theorem of linearity of integration for indefinite integrals.

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

### Example #5

In the following expression, does the linearity of integration holds?

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

Yes.

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