Friday, January 20, 2017

integral of { (cosx)^2 / [ (cosx)^2+ 4 (sinx)^2 ] }

integral of { (cos x)^2 / [ (cos x)^2+ 4 (sin x)^2 ]  }

use trigonometric formulae to change  (sin x)^2 = 1 - (cos x)^2 so that the intergral is completely in terms of (cos x)^2 .
 Now try to write the numerator in terms of the denominator.
 introduce a (-3 ) in the numerator and denominator and add and subtract 4

split it into two terms and then two integrals

The second integral contains (cos x)^2 .

divide each term with (cos x)^2 to get (sec x)^2

use trigonometric formulae tochange (sec x)^2  = 1+ (tan x)^2  in the denominator only

use substitution   t = tan x and change the limits.






formulae on integration
 
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

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Thursday, January 19, 2017

integral using substitution and then integration by parts

integral using substitution and then integration by parts

integral of [1 / (x^4)][sqrt( 1 + (x^2))][ log( 1 + (x^2)) - 2log( x)]

first simplify using property of logarithms

take x^2 common from the sqrt term obtain [1+1/(x^2)] and try to get the same term inside the log expression

cancel off the x to get 1 /[x^3]   then use substitution

then use integration by parts with log(t) as the first function



formulae on integration
 
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

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integral of sqrt{[1-sqrt(x)] / [1+sqrt(x)]}

integral of sqrt{[1-sqrt(x)] / [1+sqrt(x)]}

use the substitution sqrt(x) = cost

use trigonometric formulae to simplify [1-cos t]  and [ 1+ cos t], sint using the half angle formulae

cancel off the common factors

simplify then again use trigonometric formulae to change the square terms to first degree expressions before integrating.

again use trigonometric formulae to change the variable back to x



trigonometric identities 

formulae on integration
 
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

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Wednesday, January 18, 2017

integral of { [ arcsin(sqrt(x)) - arccos(sqrt(x)) ] / [ arcsin(sqrt(x)) + arccos(sqrt(x))] }

integral of { [ arcsin(sqrt(x)) - arccos(sqrt(x)) ] / [ arcsin(sqrt(x)) + arccos(sqrt(x))] }

x belongs to [0,1] 

use the result that arcsin(sqrt(x)) + arccos(sqrt(x))] = [pi / 2]
to get rid of arccos(sqrt(x)) and write the integral completely in terms of  arcsin(sqrt(x))

use a substitution to  change the arc sine function to a function involving sine function

use integration by parts to handle the new integral.






formulae on integration
 
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

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There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work

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


integral of 1 / [cos(x+a)cos(x+b)] by manipulating the angle in the numerator in terms of the angles of the denominator

introduce a  term sin(a-b) in the numerator by multiplying the numerator and denominator with it.

Then express the angle  (a-b)  in terms of the denominator(x+a) and (x+b)
using (a-b) = (x+a) - (x+b)

use trigonometric formula for expanding the numerator and then split the numerator

then simplify and integrate the two terms separately.

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

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Tuesday, January 17, 2017

integral of 1 / 6{ [ x^(1/2)+ x^(1/3) ] } using substitution

integral of 1 / 6{ [ x^(1/2)+ x^(1/3) ] } using substitution

here the lcm of 2 and 3 is 6


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

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There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work

integral of 1 / { x*sqrt[ax-x^2]} using substitution

 integral of 1 / { x*sqrt[ax-x^2]} using substitution

method is put x = (a/t) ; dx = [-a / (t^2) ] dt

replace x and dx in the given equation and simplify as shown below

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

disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work