Integration is the process of calculating the integral values. In mathematics, it is defined as the process of summing up the values of different functions. The parts are summed up together to create a definite value. Unlike differentiation, integration is the summation of two or more values. Integration is considered to be the invoice process of differentiation.
Therefore, integration can also be termed as an antiderivative where the process of calculating a derivative of a function is reversed. For example, The derivative of cos(x) is – sin(x). Therefore, -sin(x) is termed as the derivative of cos(x) and cos(x) is the antiderivative of -sin(x). There is a dedicated integration formula that helps with calculating integration.
∫ x^n dx = ((xn+1)/(n+1))+C where C is a constant and n cannot be 1
∫ 1 dx = x + C
∫ sin x dx = – cos x + C
The concept comes in handy in the case of addition where the value may reach infinity. The integration methods help with the easy and hassle-free calculation of addition problems. Integration is mainly used for the calculation of the values of volume and area. ∫ is the sign used for anti-derivative or integration. Integration represented as ∫ f(x) dx is F(x) + C where C is the consonant.
Integrals and their types:
Integrals in maths can be broadly categorized into two parts. These are listed below:
- Definite integral:
The definite integral is defined as the integral value which has an upper and lower limit. The value of x i.e. the variable is represented on the axis between the upper and lower limit.
- Indefinite Integral:
On the other hand, indefinite integrals are defined which do not contain any upper and lower limits. It is simply represented as follows ∫f(x)dx = F(x) + C.
For example, ∫x^4 dx can be calculated as follows:
∫f(x)dx = ∫x^4 dx
= x^5/5 + C
Derivative and its formulas:
A derivative on the other hand is defined as the change in the value of a function with respect to a variable that is independent. The rate of change of one variable with respect to another value can be easily calculated through the use of a derivative. In mathematics, the calculation of derivatives is termed differentiation. A derivative of a function y can be denoted as dy/dx.
dy is the rate of change in y and dx is the rate of change in x. Here y is the function and differentiation or derivative helps in the calculation of the rate of change in the value of with respect to x. Moreover, y is the dependent variable that is based on the rate of change in x which is an independent variable.
- derivative or d/dx of C = 0 where C is a constant
- d/dx (x) = 1
- d/dx(x*n) = n(x)^n-1
- d/dx log x = 1/x
Derivatives can be classified into two types:
- First Order derivative:
The First Order derivative gives an idea about the direction of function. The increase and decrease in the function’s value can be identified through the first-order derivative. The slope of the tangent line can give the true direction of the first-order derivative.
- Second order derivative:
The shape of a graph of a given function can be identified through a second-order derivative.
Derivative of trigonometric functions:
- d/dx sinx = cosx
- d/dx cosx = -sinx
- d/dx cotx = -cosec^2 x
- d/dx tanx = sec^2 x
A dedicated property of derivatives explains the calculation of an expression by splitting the addition, subtraction, and division signs.
Derivative of f(x) = 3x^2 + 2x – 8 can be calculated as follows:
d/dx f(x) = d/dx (3x^2 + 2x- 8)
= d/dx (3x^2) + d/dx (2x) – d/dx (8)
= 6x + 2 – 0
= 6x + 2
All the dedicated concepts and formulas of derivatives and integration can be calculated through Cuemath and its official website. The best information, learning concept videos, and help can be obtained through Cuemath.