# INTEGRATION OF SIMPLE ALGEBRAIC FUNCTIONS

CONTENT

- Integration and evaluation of definite simple Algebraic functions.
- Application of integration in calculating area under the curve.
- Use of Simpson’s rule to find area under the curve.

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**INTEGRATION AND EVALUATION OF DEFINITE SIMPLE ALGEBRAIC FUNCTIONS**

Integration is the opposite of Differentiation. It is the process of obtaining a function from its derivative. A function \(F(x)\) is an anti derivative of a given function \(F(x)\) if \(d \frac{F(x)}{dx} = f(x)\).

In general, if \(F(x)\) is any anti derivative of \(f(x)\), then the most general anti derivative of \(f(x)\) is specified by \(f(x) + c\) and we write: \(\displaystyle ∫f(x)dx + c\)

The symbol \(∫\) is called an integral sign and \(\displaystyle ∫f(x)dx\) is called the indefinite integral. The arbitrary constant \(c\) is called the constant of integration, and the function \(F(x)\) is called the integral.

For example, \(F(x) = x^4 + c\) is an anti derivative of \(f(x) = 4x^3\) because \(F’(x) = \frac{dx^4}{dx} = 4x^3 = f(x)\).

In general, if \(n ≠ -1\), then an anti derivative of \(f(x) = x^n\) is \(F (x) = \frac{x^{n + 1}}{n + 1} + C\)

To integrate a power of \(x\) (apart from power \(n = -1\), increase the power of \(x\) by \(1\) (one) and divide by the new power.

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