mixture: April 2009

Thursday, April 30, 2009

manipulation of numerator in terms of the denominator

Integration by manipulation of numerator in terms of the denominator

Integration formulae

(px+q) / (ax² + bx +c) , (px+q) /sqrt(ax² + bx +c) ,

(px² + qx +r) /(ax² + bx +c) type

(psinx+qcosx ) / (a sinx + b cosx )

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for

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you might then have to restore to completion of squares in some cases after the manipulation
in some cases the manipulation can be done through inspection

Examples

* ∫ { (4x - 7) / (x² + x +1)} dx
integration of linear expression in numerator mixed with quadratic expression

* ∫ { cosx/ [ sinx +cosx] } dx
integral of [ cos x ] / [sin x + cos x ]

*not the above types but still an example of manupulating the angle in the numerator in terms of the angles in the denominator

integral of 1 / [cos(x+a)cos(x+b)]

*integral of { (cos x)^2 / [ (cos x)^2+ 4 (sin x)^2 ] }
answer and a little explanation of integral of { (cos x)^2 / [ (cos x)^2+ 4 (sin x)^2 ] }

PAGE 1 BASIC INTEGRATION

PAGE 2 INTEGRATION BY SUBSTITUTION
PAGE 3 INTEGRATION BY COMPLETION OF SQUARES

PAGE 4 INTEGRATION BY PARTS

PAGE 5 INTEGRATION BY MANIPULATION OF NUMERATOR IN TERMS OF DENOMINATOR

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integration of some products using by parts

Integration by parts

Integration formulae

If an integral is a product of the form uv

usually u is chosen so that successive derivatives of u will become zero after a finite number of steps
provided v has a nice integral or you can make the choice using ILATE rule
inverse , log, algebraic, trigonometric, exponential

for eg. for integral of [x² cosx]

u = x² , v = cosx

but for integral of [x² ln(x)] , you should make it into integral of [ ln(x) * x²]
so that u = ln(x) , v = x²

for integral of lnx, make it into integral of [lnx * 1] with u = lnx , v = 1

in some integrals, like integral of [ (e^x) cosx ], take the integral as I, apply integration by parts a couple of times or so, and then obtain the original integral I again, then rearrange and extract the value of I.

also note the type

here split into two integrals
use integration by parts once on integral of [f(x)exp(x)] taking f(x) as the first function
this will cancel off the integral of [f '(x)exp(x)] leaving you with exp(x) * f(x ) + C which is the required answer

Examples on integration by parts

* integral of ln(x)
answer and some explanation

*integral of arc(cosx)
answer and some explanation

* integral of { [ arcsin(sqrt(x)) - arccos(sqrt(x)) ] / [ arcsin(sqrt(x)) + arccos(sqrt(x))] }
explanation and answer to the integral using integration by parts is available here

*integral of [1 / (x^4)][sqrt( 1 + (x^2))][ log( 1 + (x^2)) - 2log( x)]
explanation and answer using substitution and then integration by parts is available here

*integral of [e^2x]cos3x using integration by parts
explanation and answer of integral of [e^2x]cos3x using integration by parts

PAGE 1 BASIC INTEGRATION
PAGE 2 INTEGRATION BY SUBSTITUTIONPAGE 3 INTEGRATION BY COMPLETION OF SQUARES
PAGE 4 INTEGRATION BY PARTS
PAGE 5 INTEGRATION BY MANIPULATION OF NUMERATOR IN TERMS OF DENOMINATOR

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mode of the binomial distribution

Mode of the binomial distribution

A random variable X is said to follow binomial distribution with parameters n and p if its pdf is given by

formulae for factorials and combinations

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Website Analytics and Website Statistics by NextSTAT

some things to do with an old computer

some interesting things to do with your old computer

1. Converting the old computer into a useful multimedia player :

first of al check if your old computer runs winamp or any other audio playback software. It is usually better to use older versions of winamp which can usually be found at (oldapps.com) at (oldversion.com) . For better quality playback , the soundcard of the computer can be connected to the aux input of stereo players.Using a tv-tuner card, the old computer can also be used for watching tv .

2.Using the computer as a firewall,proxy server etc :

You can change your old computer into a firewall by using software like smoothwall (smoothwall.org). You can also use it to share a single internet connection with more than one computer by using proxy server software like wingate (www.wingate.com) , freeproxy (handcraftedsoftware.org) etc on the old computer. If you are interested in helping people circumvent censorship and have a static ip address you can also use the old computer for running censorship circumventing software ( www.peacefire.org ) on the old computer (but you have to be careful of spammers and other legal problems that may arise).

3. experiment by installing win95/98, linux or dos on the old computer .

The old computer can be used to run old software, games which might not run on new operating systems. If you wanted to experiment by installing linux or other such operating systems, the old computer will be a nice tool for that.

4. changing the old computer into a file server or printer server.

you can use the old computer as a server to store commonly used documents and mp3s. This makes it easier to backup those files. Instead of many printers connected with different computers, you can set up a network so that printing takes place on printers connected to the centralised old computer. This is useful especially if you routinely print hundreds of pages. Some of the older dot matrix printers can be connected only to serial ports.(some older accounting software write only to such dot matrix printers). The old computer (usually will have serial ports) can be used to connect such old printers .

5. Study hardware of computers

Most people will be curious about the internals of a computer, but may not be courageous to open up a new branded pc. The old computer is a nice tool to practice assembling a computer by yourself. Very good free tutorials are available on the internet.

If you are not interested in such things, you could think of donating the old computer to the needy or you could recycle the old computer. Do not throw it out as trash since the computer usually contains a lot of toxic material like lead, antimony, arsenic, selenium, mercury etc. (news.bbc.co.uk) (svtc.etoxics.org) . Be careful and try to permanently erase the contents of the old hard disk using software like eraser (sourceforge.net) , if you are giving away /donating your old computer . An article about the dangers of improper erasure of data is available here.

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integration by completion of squares

Integration by completion of squares

Integration formulae

If an integral is of

sqrt(ax² + bx +c) , 1 /sqrt(ax² + bx +c) ,1 /(ax² + bx +c) type

first complete the square

ax² + bx +c = a[ x² + (b/a)x + (c/a) ] =a[ ( x +(b/2a) )² + (c/a) -(b²/4a²) ]

remember to make the coefficient of x² unity before starting
Examples on integration by completion of squares

* integral of [ 1 / (x² -x+1)]
integral using completion of squares method

*integral of 1 / sqrt[4 + 3x - x² ]
completion of squares inside a square root

PAGE 1 BASIC INTEGRATION
PAGE 2 INTEGRATION BY SUBSTITUTIONPAGE 3 INTEGRATION BY COMPLETION OF SQUARES
PAGE 4 INTEGRATION BY PARTS
PAGE 5 INTEGRATION BY MANIPULATION OF NUMERATOR IN TERMS OF DENOMINATOR

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Tuesday, April 28, 2009

integration substitution

substitution method for integration

Integration formulae

If an integral is not of a standard type, but can be reduced to a standard type by substitution

of the form
u = f(x)
du = f ' (x) dx take the differential on both sides if necessary

after substitution, remember that everything should be in terms of u alone

Examples on integration by substitution

* integral of x^3 / sqrt(9 - x^2)
answer and some explanation

* integral of (x^3)*sqrt[4-x^2]
answer and some explanation

* integral of 1 / { x*sqrt[ax-x^2]} using substitution
explanation of the answer to integration by substitution

*integral of 1 / 6{ [ x^(1/2)+ x^(1/3) ] } using substitution
answer with some explanation

*integral of sqrt{[1-sqrt(x)] / [1+sqrt(x)]}
explanation and answer to the problem on integration

*integral of 2*cube of (tanx) with 0 to pi/4 as limits
explanation and answer to integral of 2*cube of (tanx) with 0 to pi/4 as limits

PAGE 1 BASIC INTEGRATION

PAGE 2 INTEGRATION BY SUBSTITUTION

PAGE 3 INTEGRATION BY COMPLETION OF SQUARES

PAGE 4 INTEGRATION BY PARTS

PAGE 5 INTEGRATION BY MANIPULATION OF NUMERATOR IN TERMS OF DENOMINATOR

arc length

arc length of x= t²+1 , y= 2t-3 , 0≤t≤1

Monday, April 27, 2009

exact vaue of sin(arctan(4/3)-arccos(12/13))

exact value of sin [arctan(4/3)-arccos(12/13)]

let A = arctan(4/3)
tanA = 4/3

draw a rough right triangle to get

sinA = 4/5 ; cosA = 3/5

let B = arccos(12/13)
as before cosB = 12/13 ; sinB = 5/13

now sin [arctan(4/3)-arccos(12/13)] = sin[A-B] = sinAcosB - cosAsinB trigonometric identities
=(4/5 )(12/13) - (3/5)(5/13) = 33 / 65

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Friday, April 24, 2009

integration guide (certain topics)

free online guide to certain topics on integration for plus two, isc , cbse etc

First of all try to familiarize yourself with the formulae on integration and trigonometric identities Basic Integration
you can integrate an expression term by term
that is you can break up an integral of sum or difference of two expressions into sum or difference of two integrals
but do not do that with a product or quotient of two functionsbut integral of 2f(x) can be expressed as 2 * integral of f(x) since "2" is a constantDefinite Integrals (Integrals with limits)

trying to write expressions in the form x^n
for eg.
cube root of x can be written as x^(1/3)
1 / (x ³) can be written as x^(-3)

if you know that the integral of f(x) is g(x) + C
then the integral of f(ax+b) will be ( 1/a ) g(ax+b) + C

you can divide the numerator term by term using the denominator but not the other way.if the numerator is of higher degree than the denominator use long division

if you have a product of sine and/or cosine terms use trigonometric identities for sinAcosB etc to break them into sum or difference before integrating
you can use substitution methods or trigonometric identities to evaluate integrals with powers of sine or cosines.

Examples on basic integration

integral of sin(5x)sin(8x)
answer and some explanation

integral of (x^3 +3x +4) /sqrt(x)
answer and some explanation

integral of [ x^3 - x^2 + x -1] / (x-1)
answer and some explanation

*integral of e^(2-3x) using limit of sums
answer of integral of e^(2-3x) using limit of sums

PAGE 1 BASIC INTEGRATION
PAGE 2 INTEGRATION BY SUBSTITUTION

PAGE 3 INTEGRATION BY COMPLETION OF SQUARES
PAGE 4 INTEGRATION BY PARTS
PAGE 5 INTEGRATION BY MANIPULATION OF NUMERATOR IN TERMS OF DENOMINATOR

PAGE 6 INTEGRATION USING PARTIAL FRACTIONS
PAGE 7 INTEGRATION OF 1 / [ACOSX +BSINX +C]

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visit to trivandrum zoo 5

a leopard atop a tree
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visit to the zoo 4

a couple of deer fighting it out
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visit to trivandrum zoo 3

a giraffe with a bandage on its leg
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visit to trivandrum zoo 2

many bats enjoying their siesta
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a visit to the trivandrum zoo 1

some deer enjoying their lunch

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Tuesday, April 21, 2009

finding a perpendicular vector using the concept of dot product

finding a perpendicular vector using the concept of dot product
find a vector perpendicular to both [ -1, 3, 4 ] and [-2, -1, 3]

let [x ,y, z] be the required vector

since [x ,y, z] is perpendicular to both [ -1, 3, 4 ] and [-2, -1, 3]
-1x + 3y + 4z = 0
-2x -1y + 3z = 0

three unknowns and two equations
choose one unknown as arbitrary

put x = a

3y + 4z = a
-y +3z = 2a

solving

y = -5a / 13
z = 7a / 13

so if a = 13

x = 13 , y = -5 , z = 7

therefore [13 , -5 , 7] or any of it's scalar multiples will be the required vector
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derivative of sqrt(3x) from first principles

derivative of sqrt(3x) from first principles
let f(x) = sqrt(3x)
replace x with x+h to get f(x+h)
take the difference f(x) - f(x+h)
introduce the conjugate before taking limit as h--> 0

Friday, April 17, 2009

problem on automorphism on abelian group

If G is an abelian group ,
let f : G -->G be defined by f(x) = xֿ¹ show that f is an automorphism on G

volume using integral calculus

Volume of the solid obtained by revolving the shaded region about the x-axis is

solved problems on finding volume of solid of revolution using integration

volume of the solid obtained by revolving the ellipse ( x² / a²) + ( y² / b²) = 1 about the x axis

volume of the solid obtained by revolving the cardioid r = a(1 + cosθ ) about the initial line

volume of the spindle shaped solid obtained by revolving the
hypocycloid x^(2/3) + y^(2/3) =a^(2/3) about the x axis.

Thursday, April 16, 2009

calculating area using integration

calculation of area using integration (calculus method)

The area of the region bounded by y = f(x) , the x-axis , the horizontal lines x = a , x = b is given by

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The area of the region bounded by x = g(y) , the y-axis , the vertical lines y = c , y = d is given by

some problems on finding area using integration

area under one arch of the cycloid x = a(t-sint) , y =a(1-cost) ----------problem on area by integration

area between y²= x and x²=y -------------area between y²= x and x²=y

find the area of the region {(x,y) / x ² +y ² <= 1 <= x+y}
answer: (pi/4) - (1/2) more explanation on this area question

area bounded by y = x ² +2 and y = 3x ----------area between two curves
area of the cardioid r = a(1 - cosθ ) in polar form ----------- area in polar form

show that e^(ix) = cosx +i sinx using power series

show that e^(ix) = cosx + i sinx using power series

Wednesday, April 15, 2009

exact values of certain trigonometric ratio

exact values of certain trigonometric ratio for 7½ ° , 15° , 18° ,22½° , 36° , 54°,72°,75°

proof of exact value of tan[22½ °] or tan (pi / 8 )

exact value of tan15° without using calculator

Monday, April 13, 2009

centre and radius of a circle

find the centre and radius of 3x^2 + 12x+ 3y^2 -5y - 2 = 0

first divide by 3, in order to make the coefficient of x^2 and y^2 unity

then compare with general equation some formulae on circle / or complete squares

Monday, April 6, 2009

bsnl directory

bsnl website

The online bsnl directory is available at bsnl.co.in

In particular that for kerala is available here

india post website

speed post tracking for india post

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auto ad

Thursday, April 30, 2009

(px+q) / (ax² + bx +c) , (px+q) /sqrt(ax² + bx +c) ,

(px² + qx +r) /(ax² + bx +c) type

(psinx+qcosx ) / (a sinx + b cosx )

If an integral is a product of the form uv

If an integral is of

sqrt(ax² + bx +c) , 1 /sqrt(ax² + bx +c) ,1 /(ax² + bx +c) type

ax² + bx +c = a[ x² + (b/a)x + (c/a) ] =a[ ( x +(b/2a) )² + (c/a) -(b²/4a²) ]

Tuesday, April 28, 2009

If an integral is not of a standard type, but can be reduced to a standard type by substitution

Monday, April 27, 2009

Friday, April 24, 2009

Tuesday, April 21, 2009

Friday, April 17, 2009

Thursday, April 16, 2009

Wednesday, April 15, 2009

Monday, April 13, 2009

Monday, April 6, 2009

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