Calculus: Rules of Integration

Calculus Integration Rules
Following are the Rules of Integration where a, b, c, n are some constants and u=f(x);v=g(x) and w=h(x)
General Integration Rules
1.Integral [a dx]= ax +c
2. Integral [a f(x) dx]= a Integral [f(x) dx]
3.Integral [x^n dx]= x^(n+1)/(n+1) +c
4.Integral [f(x)+g(x)+h(x)]dx = Integral[f(x)dx]+ Integral[g(x)dx]+ Integral[h(x)dx]
5. Integral[f(x)-g(x)-h(x)]dx = Integral[f(x)dx] – Integral[g(x)dx] – Integral[h(x)dx]
6. Integration by parts: Integral [u dv] = uv – Integral [v du]
7.Integral[F(u)dx] = Integral [F(u)/u’] du
8. Integral[1/x dx] =ln|x| +c
9. Integral [1/(x^2+a^2)]dx =1/a tan^-1[x/a] +c
10.Integral [1/(x^2-a^2)]dx = [1/2a ]ln |x-a/x+a| +c

Integral Rules of Exponential Functions
1.Integral [e^x dx]= e^x +c
2. Integral [a^x dx]= a^x/ln a +c
3.Integral [ln x dx]= x(ln x -1) +c
4.Integral[log base a of x]dx= (x/ln a)(ln x -1) +c
5. Integral[x e^(ax)]dx= [e^(ax)/a^2](ax-1) +c
6. Integral[e^(ax)/x] dx = ln|x| + summation(i=1 to infinity) [(ax)^i/i.i!] +c
7.Integral[x^2 e^(ax)]dx= e^(ax)[(x^2/a – 2x/a^2 + 2/a^3)] +c
8. Integral[x^n e^(ax)]dx = (1/a)x^n e^(ax) – (n/a)Integral [x^(n-1) e^(ax)]dx
9.Integral[e^(ax)/x^n] dx = [1/(n-1)][- e^(ax)/x^(n-1) + a Integral e^(ax)/x^(n-1)]dx
10.Integral[x^n ln x] dx= [x^(n+1)]/(n+1)^2 [(n+1)lnx – 1] + c

Integration Rules of Trigonometric Functions
1.Integral [sin x dx]=  - cos x +c
2. Integral [cos x dx]= sin x +c
3. Integral [tan x dx]= ln |sec x|+ c
4. Integral [cot x dx]= ln|sin x| + c
5. Integral [sec^2(x) dx]= tan x +c
6. Integral [csc^2(x) dx]= - cot x +c
7. Integral [tan^2(x) dx]= tan x – x +c
8. Integral [cot^2(x) dx]= cot x – x +c
9.Integral [sec x tan x dx]= sec x +c
10. Integral [csc x cot x dx] = - csc x +c
11. Integral [sec x dx]= ln |sec x +tan x| +c
12.Integral [cos^2(x) dx] = x/2 + ¼(sin 2x) +c
13. Integral[sin ^n(x) dx]= (-1/n)sin^(n-1) x cos x + (n-1)/n .Integral [sin^(n-2) x dx]
14. Integral [cos^n(x)dx]= (1/n)cos^(n-1) x sin x + (n-1)/n. Integral[cos^(n-2) dx]

Integration Rules of Hyperbolic Functions
1.Integral[sinh x dx] = cosh x +c
2.Integral[cosh x dx]= sinh x +c
3. Integral[tanh x dx]= ln cosh x +c
4. Integral[coth x dx]= ln |sinh x| + c
5. Integral[sech x dx] = sin^-1[tanh x] +c
6. Integral[csch x dx] = ln tanh (x/2) +c
7. Integral[sinh^2(x)dx] = (sinh 2x)/4 – (x/2) + c
8. Integral[cosh^2(x)dx]= (sinh 2x)/4 + (x/2) +c
9. Integral[sech^2 (x) dx]= tanh x +c
10. Integral[csh^2(x)dx]= -coth x +c
11. Integral[tanh^2(x) dx]= x – tanh x +c
12. Integral[coth^2(x)dx]= x – coth x +c
13.Integral[sechx tanh x] dx =  - sech x +c
14. Integral[csch x coth x]dx = -csch x +c

Derivatives of Exponential functions of e

Derivative is the rate of change at a point which gives the slope of the curve at that point. When the given equation is y=f(x), the derivative is written as dy/dx or d[f(x)]/dx.  To find the derivatives of exponential functions, let us take a quick look at them. Exponential functions are the functions written in the form y = b^x, where b is a positive number that does not equal 1 and x is any real number.  They have a constant base and the exponent is a variable. The most important exponential function is e as the base, which is an irrational number. The function is written as, e(x) and is called the natural exponential function.  Now that we learnt about the natural exponential function e^x, let us learn more about the Derivatives of E.

The natural exponential function is remarkable and so are its derivatives. Let us first find the derivative of E, where E is f(x)=e^x :
As per the definition of derivatives, we get,
d[f(x)]/dx = lim(delta(x)?0) e^[x+delta(x)-e^x]/delta(x)
=lim(delta(x)?0) [e^xe^delta(x)- e^x]/delta(x)
=lim(delta(x)?0) e^x[e^delta(x)-1]/delta(x)
=lim(delta(x)?0) e^x[1+delta(x)-1]/delta(x)
= lim(delta(x)?0)e^xdelta(x)/delta(x)
=e^x
If f(x) = e^x then f’(x) = e^x. This means that slope is the same as the given function value or value of y for all the points on the graph. The other Derivatives of E or derivatives of e^x are as given below:
If u is a function of x, the derivative of an expression in the form e^u can be obtained and is given by d(e^u)/dx = e^u. du/dx

If an exponential function with base b is given, then the derivative of that expression is given by
d(b^u)/dx = b^u.ln b.du/dx

Let us take an example, derivative of E 2 which is derivative of e^x where x=2.  At this point x=2, the value of y=e^x  is approximately 7.39. We know that the derivative of e^x is e^x. So, the slope of the tangent, that is the derivative of e^2 at x=2 is also 7.39 approximately.

Derivative of E 2x will be the derivative of e^2x. To find the derivative of this exponential function, let us take y= E 2 or y= e^2x where u=2x. Using the chain rule, we get  dy/du = de^u/du , where du/dx equals 2.  So, d/dx of [e^2x] is (e^u). du/dx = 2. e^u , substituting u=2x, the derivative of e^2x is 2e^2x

Derivative of E 3x will be the derivative of e^3x. Using d(e^u)/dx = e^u. du/dx where u=3x, we get, e^3x. du/dx which will be 3e^3x as du/dx = 3

Law of Cosines Explained

Trigonometry Law of cosines:
Trigonometry is a field of study relating the angles and sides of a triangle. However, the fundamental ratios are derived easily from a right-angled triangle and are as identified Pythagorean ratios, yet the definite correspondence between the sides and angles can be established using the law of cosines. It is the relation between the sides and cosine of angle.


Convention:
All the angles are depicts upper case letters and sides are depicted by lower case letters. The side opposite to a vertex is represented by the corresponding lower case letter. In a triangle ABC, the side AB= c, BC = a, and AC = b.


Trigonometry law of cosines:
a^2= b^2+c^2 - 2bc cos (A)
b^2 = a^2 +c^2 – 2ac cos (B)
c^2 = a^2 + b^2 – 2ab cos (C)


Law of cosines Example Problems: 
For example, let us consider a triangle ABC, in which a= 3, b= 4 and c= 5.
To evaluate the angle C,
Substitute the values of a, b and c in
c^2 = a^2 + b^2 – 2ab cos (C)
5^2 = 3^2 + 4^2 – 2(3) (4) cos (C)
25= 9+16 – 24 cos(C)
25 = 25 – 24 cos(C)
Solving for cos (C), we get cos(C) =0. Hence, C = 90?.


Prove Law of Cosines:
We always rely on the principles of geometrical principles to prove the laws in trigonometry. To prove the law of cosines we use Pythagorean principle.

 In the above figure, CP is perpendicular to AC extended to P. Hence, BP = a sin(C) and CP = a cos(C).
Applying Pythagoras theorem in the right-angled triangle APB we get,
(AP)^2 + (BP)^2 = (AB)^2
(b – a sin(c))^2 + (a sin(C))^2 = c^2
Expanding        b^2 -2ab cos(C) + b^2 cos2(C) + b^2sin2(C) = c^2
b^2 +a^2 – 2ab cos(C) = c^2


Law of Cosines Problems
Law of cosines, in Physical sciences and technology, has very wide applications. The law of cosine gives us the magnitude of the difference of two vectors acting at an angle. For example, to evaluate the magnitude of the difference of two vectors of magnitude 100 units each acting at an angle of 120?, we get
c^2= 1002+1002 – 2(100) (100) cos (120)
c^2= 1002+1002 -1002
Solving for c, we get c= 100 units.

Derive Law of Cosines
Though there are so many methods by which one can prove the law of cosines, we stick to the use of Pythagorean principles.

In the above figure we have, when CP is perpendicular to AB,
c= a cos (B) + b cos (A)
c^2= ac cos (B) +bc cos (A)
Similarly
b^2=bc cos (A) + ba cos(C)
a^2= ab cos(C) + ac cos (B)
Adding the above two equations we have
b^2 + a^2 = ac cos (B) + bc cos (A) + 2 ab cos(C)
Comparing the above equation with (1), we get
b^2 + a^2 = c^2 + 2ab cos(C)
b^2 + a^2 -  2ab cos(C) = c^2

Derivatives and Graphs of Exponential and Logarithmic Functions

Exponential and Logarithmic Functions
The Logarithmic function with base b is a function, y = logb x. Here b is greater than zero and the function x is defined for all x greater than zero. An Exponential function with base b is a function, y=bx, defined for every real number.
Inverse Function: To find an inverse function (f-1), we need to interchange x and y and then solve for y. Example: f -1( x)  of 2x +1 will be, y=2x+1 (interchange x and y and solve for y)
x =2y+1
y = (x-1)/2 = f-1(x)
The Exponential functions and Logarithmic functions are inverse functions, that is, for any base b, the functions f(x) = logb x, g(x) = bx  are inverses.
Example: let f(x) =ln x and g(x) = ex then f and g satisfy the inverse functions. f(g(x) = ln ex=  x and g(f(x) = eln x= x, f(g(x) = g(f(x) and hence the functions f(x) and g(x) are inverses

Derivatives of Logarithmic and Exponential Functions
The most common exponential and logarithmic functions are natural exponent function ex, and the natural logarithm function, ln(x). The derivatives of Exponential and Logarithmic Functions are:
Exponential Functions derivative: d/dx (ex) = ex   d/dx(ax) =ax ln a
Logarithmic Functions derivative: d/dx(ln x) =1/x    d/dx(loga x)= 1/x ln a
Example: Derivative of f(x) = e3x+2  is given by d/dx (e3x+2) = e3x+2. 3 = 3e3x+2
   Derivative of f(x) = ln (3x+2) is given by d/dx[ln (3x+2)] = [1/(3x+2)]. 2 = 2/(3x+2)

Graphing Exponential and Logarithmic Functions
Exponential functions play a large role in real life. From science to money, graphing these exponential functions provide a visual representations to many applications in real life. Graphs of Exponential and Logarithmic functions using examples are as follows,
Let us now graph an exponential function, f(x) = 2x. First we evaluate f(x) using the integers -3, -2, -1, 0, 1,2,3 and tabulate the values.
x   -3          -2        -1        0 1 2        3
f(x)    1/8 1/4 1/2 1 2 4 8


(x,y)   (-3,0.125)      (-2, 0.25)   (-1,0.5)   (0,1)       (1,2)         (2,4)       (3,8)
Once we get the ordered pairs (x,y) plot the points which gives us the graph of the exponential function f(x)=2x.

Graphing Logarithmic Functions: There are several ways to graph logarithmic functions. The easiest way to graph them is to re-write them in exponential form.
Example:  Graph the logarithmic function, f(x) = log5 x. Re-writing f(x) = y =log5 x in exponential form we get x = 5y, choose values for y and then compute corresponding values for x.  Tabulating the values,
y   -1         0 1        2
x = 5^y 1/5         1             5       25
(x,y) (0.2,-1)     (1,0)           (5, 1)      (25, 2)
Plot the (x,y) values. The graph we get is the graph of the logarithmic function, f(x)= log5 x.

Natural Numbers

Number theory : The set of it integers and its properties are at  the  root of all mathematical disciplines. In fact , it is impossible to do mathematics without making use of integers in some form or another. Number theory which involves the study of integers  itself , is a rich and fascinating branch o mathematics .many volume have been written on this subject and some of the best mathematician in history have devoted much of their time to the study of number theory.

We can subdivide the number theory as follow :
(i) Combinatorial Number Theory
(ii) Algebraic Number Theory
(iii) Analytic Number Theory
(iv) Transcendental number theory
(v) Geometric number theory
(vi) Computational number theory

What is a Natural Number?
Introduction To natural number : Since our childhood we are using numbers 1 , 2, 3, 4,………………………..t count and calculate. For example 3 banana , 5 apples , 7 mangoes , 2 books etc. here banana , apples , mangoes are objects whereas three , five , seven , two etc indicates about the quantities of theses objects .

To define natural numbers we might put these in this way, as when we count objects in groups of objects , we start counting from one and then go on to two, three , four etc . there is a  natural way of counting  objects .Hence 1 , 2 , 3 , 4 , …………………are called natural numbers .In fact number from 1 to 1crore are all natural numbers.Let us see what  is  a natural numbers ?We start counting from1  , so 1 is the first natural number , if we add1 to the first natural number , then we get 2  the second natural number/.by adding 1 to any natural number , then we get 2 , the second natural number. In fact adding 1 to  any natural number , we get the next natural number let us take few  examples of natural numbers  , 1000 is the natural number next to 999 , 10001 is the natural number, next to 10000 and so on .Thus if we think of any natural number , there is always a natural number next to it .Consequently there is no last  or greatest natural number consequently there is no last or greatest natural number. now in simple word , let  us define natural numbers: natural numbers are number from 1 onwards, ie , 1 , 2  3  4 , 5 , 6 ……………………..and are used for counting

Properties of natural numbers :Following are some properties of natural number (i) The first and smallest natural number is 1.(ii) Every natural number (except 1) can be obtained by adding to 1 to the previous natural number(iii)For the natural number 1, there is no previous natural number (iv)There is no last or greatest natural number (v)We cannot complete the counting of all natural number .

Ratios and Scale drawings

A ratio is a comparison of one thing with another.  It shows the relative size of two or more quantities. The order of the ratio is very important. a:b is not equal to b:a. Now let us find how to do ratios? A ratio can be expresses in three different ways:-

1. Fractional notation : 2/5
2. Odds notation: 2:5
3. Using the word “to” : 2 to 5

How to find ratios? Ratios can be solved by reducing them.  We can multiply or divide both the terms of ratio by the same number; it makes no change to the ratio. So this clearly answers how to solve ratios?

For example: - In a class of 20 students, 5 are girls and 15 are boys. What is the ratio of girls to boys?
Solution: - There are 5 girls and 15 boys.
The ratio of girls to boys will be 5:15
We can reduce this fraction by dividing both the terms by number 5
The ratio will be 1:3
Let us learn about ratio word problems now. Ratio word problems are problems that require use of ratios to relate the different quantities.
Following points should be kept in mind while solving ratio word problems:-

• Convert all the quantities to same units if required
• Write the quantities in the ratio as a fraction
• Remember to keep same quantities in numerator and denominator

Example:- There are blue and red ball in a bag, the ratio of blue balls to red balls is 5:6. If the bag contains 60 red balls, how many blue balls are there?
Solution: -
Let x be blue balls
Red/Green = 5/6 = x/60
X = 50 balls.
Let us learn about scale drawings. If any ratio is expressed in 1:n  form, then n is called the scale factor. It is not possible to draw on paper the exact size of real- life objects. Therefore we make use of scale drawing to draw such figures like a car, a building or any map.

Example: -A figure has a scale of 1:10 that means anything drawn with the size of “1” would have a size of “10” in real world.

How to Convert Fractions to Decimals

Let's learn about How to Convert Fractions to Decimals.

In order to Convert Fractions to Decimals, the proper or improper fraction has to undergo division. The numerator has to be divided by the denominator and the resultant will be a decimal value. Below is an example:
Example:
5/2
= 2.5

For more help, get it from an expert free math tutor. Next time i will help you with the concept of solving equations with decimals.

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