Let's consider the function f(x) = x² +3 its derivative is f '(x) = 2x. The function f '(x) = 2x is the derivative of the function f(x) = x² + 3. The function f(x) = x² +3 is the antiderivative of the function
f '(x) = 2x. The antiderivative is the function from which the derivative was calculated.
If we consider the function f(x) = 2x it can be considered as the derivative of another function F. This function which is F(x) = x² + 3 is the antiderivative of the function f. Actually, there are several functions whose derivative is f(x) = 2x. Some examples of these functions are F(x) = x²+1, F(x) = x²-3, F(x) = x² + 6. There is a bunch of functions whose derivative is f '(x) = 2x. We can't list them all. For this reason we represent all them by the function F(x) = x²+C.
Notation and notion of indefinite integral
The process of finding the antiderivative of a function is called antidifferentiation or integration. If F is the antiderivative of a function f we write ∫f(x)dx = F(x) +C. The left side of this equation is called integral of f(x). The function f(x) is called the integrand. The letter C is the constant of integration. The symbol dx means that the function is integrated with respect to x.
Formula to calculate the antiderivative of the function power
Let's retake the function f(x) = 2x. Its antiderivative is F(x) = x² + C or F(x) = 2/2x² + C. Let's consider another function g(x) = 3x. Its antiderivative can be G(x) = 3/2x²+C. For the function h(x) = x² its antiderivative is H(x) = 1/3x³+C. For the function i(x) =5 x³ its antiderivative is I(x) = 5/4x⁴ + C
By observing each of these functions and their antiderivative, two points draw our attention.
1) The exponent of x in the antiderivative function is one degree more than the exponent of x in the given function.
2) The coefficient of x in the antiderivative function is obtained by dividing the coefficient of x in the given function by the exponent of x in the antiderivative function.
Let's come back to the examples above. In the function antiderivative F(x) the exponent of x is 2 and the exponent of x in the function f is 1. The coefficient of x 2/2 in the function F(x) is equal to the coefficient 2 of x in the function f(x) divided by the exponent 2 of x in the antiderivative function.
In the function antiderivative G(x) the exponent of x is 2 and the exponent of x in the function g(x) is 1. The coefficient 3/2 of x in the function G(x) is equal to the coefficient 3 of x in the function f(x) divided by the exponent 2 of x in the function G(x). The same is true for the functions H(x) and I(x).
From these observations, we can deduct a rule. In order to find the antiderivative of f((x) = xⁿ add 1 to the exponent and divide the coefficient by the new exponent,
The formula that generalizes this procedure is as follows:
If f(x) = kxⁿ its derivative is given by F(x) = k/n+1xⁿ⁺¹ + C.
That's all for today. To learn more about Calculus visit Center for Integral Development
