Subject: Calculus

Antiderivative Indefinite Integral

We say that the inverse of addition is subtraction, the inverse for multiplication is division and the inverse of raising to powers and extracting roots are a few of the famous inverse operations. Then, what's the inverse process of finding the derivative? Any guess? Well, the inverse process of finding the derivative is integration or antidifferentiation (as its name suggest). However, not all differentiable functions has its corresponding anti-derivative since there is a class of functions in which antidifferentiation is impossible.
Nevertheless, let me lead you directly to our topic which is anti-derivative of functions. Basically, when a function f(x) is differentiated with respect to x, then its notations may become f'(x). But, when we want to bring f'(x) back to its original function f(x), then all we have to do is take the inverse process of differentiation. And that means we have to integrate or anti-differentiate f"(x) to make it f(x).

Notation

Integration is conventionally performed by indicating a symbol \int which is an elongated letter S, standing for summa (Latin for "sum" or "total"). Though Newton has a different notation for integration which is a letter placed inside a box, Leibniz's notation (which is the \int ) is the most widely used notation for integration and the one which we shall be using throughout the Calculus section.
So we now have the symbol for integration which means that the expression

\int f(x)dx

is read as " the integral of the function f(x) with respect to x", where f(x) is also called as the integrand and the dx is called the variable of integration.

Indefinite and Definite Integral

There are also two types of integral which must not be taken for granted especially when dealing with its notations. These are definite and indefinite integrals whose notations are given below.

Indefinite integral

\int f(x)dx

Definite integral

\int_{a}^{b} f(x)dx

a and b are called lower and upper limits respectively.

Important Concepts and Theorems

Now you know a lot of basic information about integration and I bet you're really good to go. But we're just getting started. Before introducing to you how to perform antidifferentiation or integration, allow me first to share to you some pretty important theorems and concepts behind integration.

Integral as an area

A function of a real variable x and an interval [a, b] of the real line, the definite integral

\int_{a}^{b} f(x)dx

is defined informally to be the net area of the region in the xy-plane bounded by the graph of , the x-axis, and the vertical lines x = a and x = b. Refer to the figure at the right.

Indefinite integral

A collection of functions which forms the set of all antiderivative of a certain function is called an indefinite integral. This means that the integral of a function can be many and is non-unique.
Since an indefinite integral is a set all anti-derivatives of function, and due to the fact that the derivative of a constant is zero, thus any constant may be added to an antiderivative and still correspond to the same integral. If we write the constant as C, then an indefinite integral when the antiderivative of f is F ,can be written as

\int f(x)dx= F(x)+C

C is called the constant of integration and is true for all indefinite integrals for the reasons stated above.

Definite integral

Unlike indefinite integrals, definite integrals can easily be determined because of the upper and lower limits appearing in the integral sign and unlike indefinite integrals, definite integral is unique. Also, it has no constant on integration C. Evaluating an indefinite integral is just finding the anti-derivative of a function. Well, that's not the case for indefinite integrals as stated in the theorem below.

Theorem

Let f be a well-behaved function such that its anti-derivative is F. Then the definite integral of f is given by

\int_{a}^{b} f(x)dx= F(b)-F(a)

where a and b are the endpoints in the interval [a, b].

Finding the Integral of a Function

Finding the derivative of any function varies from the type of the function. For example, the formula for finding the derivative of a polynomial function does not necessarily apply for trigonometric functions. While there is really no general formula for finding the integral of any given function, we can refer to tables of integrals for integral values. For beginner's sake here are some of the integrals of a few basic functions.

Basic Functions

\int dx = x + C

\int a dx = ax + C

\int x dx = \frac{1}{2}\cdot x^2 + C

\int ax dx = \frac{a}{2}\cdot x^2 + C

In general, the integral of any function with r-exponent is given by the following formula:

\int x^r dx = \frac{1}{r+1}\cdot x^{r+1} + C

Exponential and logarithmic functions

\int \frac{dx}{x} = \ln{x}+ C

\int \frac{dx}{ax} = \frac{1}{a}\ln{x}+ C

\int e^{x}dx = e^{x}+ C

\int e^{bx}dx = \frac{1}{b}e^{bx} + C

where b is a constant

\int a^{x}dx = \frac{1}{\ln{a}}\cdot a^{x} + C

where a is a constant

\int a^{bx}dx = \frac{1}{b\cdot \ln{a}}\cdot a^{bx} + C

where a and b are constants

Ttrigonometric functions

\int \sin{u} du = -\cos{u} + C

\int \cos{u} du = \sin{u} + C

\int \tan{u} du = \ln{|\sec{u}|} + C

\int \cot{u} du = \ln{|\sin{u}|} + C

\int \sec{u} du = \ln{|\sec{u}+\tan{u}|} + C

\int \csc{u} du = \ln{|\csc{u}-\cot{u}|} + C

We can always check the integrals above if these really are the corresponding integrals of the functions given above. Since integral is an inverse of differentiation as mentioned above, we can differentiate the integral and the answer must be the same to the original function. For example;

\int x dx = \frac{1}{2}\cdot x^2 + C

So let's differentiate the integral above,

D_{x}[ \frac{1}{2}\cdot x^2 + C]
= 2\cdot \frac{1}{2}\cdot x + 0]
=x

which is just the original function, isn't it? the same is true for all types of integrals. You can try on your own all the integrals above. For more integrals, just go to Table of Integrals and List of Integrals.

Example #1

Finding the area under a curve f(x) is determined by integrating f(x). What figure/shape represents the area whose curve is f(x)=c, where c is any constant, and the integration limits ranges from 0 to c.

Square

Example #2

What would be the area of the region under the curve f(x)=e^x\sin{\log{x}}+x^2\cos{e^x} when a=b?

Zero

Example #3

Finding the area under a curve f(x) is determined by integrating f(x). What figure/shape represents the area whose curve is f(x)=c, where c is any constant, and the integration limits ranges from c to 3c.

Rectangle

Example #4

Solve the following indefinite integral.

\int e^{-x}dx

\int \frac{1}{x}dx = \ln{x} + C

Example #5

Solve the following.

\int_{a}^{b} f(x) dx + f(a)

f(b)

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