Introduction to indefinite integrals

In derivatives we learn about the differentiability of a function on some interval I and if it is differentiable, how to find its unique derivative f’ at each point of I. In application of derivatives we learn that using derivative we can find the slope of the tangent at any point on the curve, we can find the rate of change of one variable with respect to the other. Now let us look at an operation that is inverse to differentiation. For example we know that the derivative of x^5 with respect to x is 5x^4. Suppose the question is like this: derivative of which function is 5x^4∫ Then it may not be that easy to find the answer. It is a question of inverse operation to differentiation.

The answer to the question: " Whose derivative is a given function f ∫ " is provided by an operation called anti derivation. It is possible that we may not get an answer to this question or we may have more than one answer. For example, (d/dx) (x^4) = 4x^3, (d/dx)(x^4 + 3) = 4x^3 and in general (d/dx)(x^4+C) = 4x^3, where c is some constant.

Definition of integration (integrals): If we can find a function g defined on the interval I such that (d/dx)(g(x)) = f(x), for all x belonging to I, then g(x) is called a primitive or anti derivative or indefinite integral of f(x). It is denoted by ∫ f(x) dx and is called indefinite integral of f(x) with respect to x. The process (operation) of finding g(x), given f(x) is called indefinite integration.

Thus the question when can we find the integral of f cannot be easily answered. There are some sufficient conditions such as, continuous functions and monotonic functions have integrals. Sin x/x is continuous, hence ∫(sin x/x)dx is defined but cannot be expressed as any known elementary function. Similarly, ∫ v(x^3+1) dx and ∫v(csc x) are defined but canoe be expressed as known elementary functions. If anti derivative of f exists, then it is called integrable function.
(1) ∫ f(x) dx means, integral of f(x) with respect to x.
(2) In ) ∫ f(x) dx, f(x) is called the integrand.
(3) In ) ∫ f(x) dx, ∫ …. dx indicates the process of integration with respect to x.

For evaluating indefinite integrals we use the following standard table of indefinite integrals:

Calculus - Limits and Continuity

The concept of Limitlays the foundation for the popular branch of Mathematics called Calculus. Calculusinvolves the analysis of functions and their behaviour. To study the behaviour of functions one needs to have a good hold on the fundamental concepts of Limit and Continuity. Calculus Limits and Continuity is the language of science and engineering.Limits and Continuity in Calculus have led to the development of ideas like derivative, integration etc.

Consider a function f(x):R->R and ‘a’ is any point in the domain of the function, limit of the function f(x) exists at x = a, if f(x) = f(a), where x is arbitrarily close to a. The idea is that as we approach the point x = a on the real line, f(x) approaches f (a). The limiting value of the function when x is very close to ‘a’ is ‘f (a)’. In the language of mathematics it is commented as “lim x->a, f(x) =f(a)”.

Continuity is another important concept which is based on the concept of limits. The function f(x) is said to be continuous at x = a, if the limit of the function at x = a on approaching the point x = a from both sides is equal to the value of the function at x = a. Precisely this can be stated as – if  lim h->0, f(a+h) = f(a-h)=f(a), then f(x) is continuous at x = a. Mathematically this can be stated as – a function is said to be continuous at a point if the right hand limit and the left hand limit of the function at that point is equal to the value of the function at that point. Geometrically speaking, a continuousfunction is the one which can be sketched on paper without lifting the pen even once.


 Properties of Continuous Functions:There are two important results for continuous functions which are stated in the form of theorems-
• Intermediate value theorem: The intermediate value theoremstates that if f is a real valued continuousfunctionon the closed interval [a, b] andt is any number between f(a) and f(b), then there exists a numberc in [a, b] such that f(c) = t.Example: The height of a child increases from 1 m to 1.7 m between the ages of eight and sixteen years, then, at some point of timebetween eight and sixteen years of age, the child must have had a height of 1.5 m.
• Extreme value theorem: The extreme value theorem states that if f is continuousreal valued function on [a,b], then f has a maximumvalue in [a,b], i.e. there exists some c in [a,b] such that f(c) >= f(x)  for all x in  [a,b]. The same holds for the minimum of f(x). For example, consider the function sin(x) where x lies between [0,2*pi]. This function attains at maximum at x = pi/2 and a minimum at x = 3*pi/2.

Derivatives of Exponential Functions with Trignometric function power

What are exponential functions? A mathematical function which is in the form, f(x) = a^x is called an exponential function, here x is a variable and a is a constant which is the base of the function (a is greater than zero but does not equal one). The most commonly used exponential function is e which is the natural exponential function which is denoted as e^x. Some definitions of e are
• e=lim(n?inifinity)[1+1/n]^n
• Lim(h?0)[e^h-1]/h=1, where e is a unique positive number
• e=summation(n=0 to infinity)[1/n(factorial)]
We know that derivative is defined as, f’(x) = lim(h?0)[f(x+h) – f(x)]/h
Using the above, we can find the derivative of the natural exponential function f(x) = e^x
d[e^x]/dx = lim(h?0)[e^(h+x) – e^x]/h
    =lim(h?0)[e^h.e^x – e^x]/h  [using rule of exponents]
    =lim(h?0)e^x[e^h-1]/h            [taking e^x common]
    =e^x lim(h?0)[e^h-1]/h
    =e^x .1
    = e^x
As we observe, e^x is its own derivative. Let us find the derivative of the exponential function with trigonometric function as the power.

E-sinx
We have, derivative of e^u is given by e^u.du/dx. So, the derivative of E-sinx written as e^-sinx would be d[e^-sinx]/dx. Taking –sinx = u which gives du/dx = -cosx, we get, e^u. du/dx which equals e^-sinx.(-cosx)
Finally we get the derivative of E-sinx as –cosx.e^-sinx

E –sinx
We have, y= e^-sinx
     y= e^u where u= -sinx, derivative of u= du/dx = -cosx
taking derivative, we get,
dy/dx= dy/du. du/dx
          = e^-sinx. [-cosx]
          = -cosx.e^-sinx

E^-sinx
Let us find the derivative of the exponential function with the trigonometric function –sinx as the power.
Using the chain rule, we get,
d[e^(-sinx)]/dx = d[e^(-sinx)/dx. d[-sinx]/dx
             = e^(-sinx). (-cosx)
             = -cosx.e^(-sinx)