Real life applications of integration
Integration is used for many reallife applications. Many derivations in physics and chemistry are done using integration. Say for example to calculate the halflife of a radioactive substance. It is also used in some parts of biology.
Integration is also used in many business related applications like: finding the cost function from marginal cost function, finding the revenue function from marginal revenue function, etc.
Integration Rules
Antidifferentiation
Integration is an inverse process of differentiation. Instead of differentiating a function, we are given the derivative of a function and asked to find its original function. Such a process is called integration or antidifferentiation. Therefore, the primary object of differential calculus is: 'given a function, to find its differential coefficient' whereas the primary object of integral calculus is its inverse, i.e., 'given the differential coefficient of a function, to find the function itself'.
Definition: Let f(x) be the given function of 'x'. The family of all antiderivatives of a function f(x) is called the indefinite integral of f with respect to x and is denoted by f(x) dx.
If φ(x) is any function such that φ'(x) = f(x), then f(x) dx = φ(x) + c, where 'c' is an arbitrary constant, called the constant of integration.
In the notation: f(x) dx, the symbol was introduced by Leibniz and is called the integral sign, the function to be integrated, i.e., f(x) is called the integrand, and 'dx' indicates that 'x' is the variable of integration.
Standard forms: We know the differential coefficients of standard functions. Therefore, below are the list of some elementary standard forms of integrals which are obtained from those known results.
Differentiation formulae 
Integration formulae 
(x) = 1 
1 dx = x + c 
(kx) = k 
k dx = kx + c 
(x^{n + 1}) = (n + 1)x^{n} 

(sin x) = cos x 
cos x dx = sin x + c 
(cos x) = – sin x 
sin x dx = – cos x + c 
(tan x) = sec^{2} x 
sec^{2} x dx = tan x + c 
(cot x) = – cosec^{2} x 
cosec^{2} x dx = – cot x + c 
(sec x) = sec x tan x 
sec x tan x dx = sec x + c 
(cosec x) = – cosec x cot x 
cosec x cot x dx = – cosec x + c 
(ln x) = 
dx = ln x + c 
(e^{x}) = e^{x} 
e^{x} dx = e^{x} + c 
(a^{x}) = (ln a)a^{x} 
a^{x} dx = + c, a > 1 
(Sin^{– 1} x) = 
dx = Sin^{– 1} x + c 
(Tan^{– 1} x) = 
dx = Tan^{– 1} x + c 
(Sec^{– 1} x) = 
dx = Sec^{– 1} x + c 



