INTEGRATION OF SIMPLE ALGEBRAIC FUNCTIONS

CONTENT

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

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.