integral of e^(2-3x) from 0 to 1 using limit of sums

integral of e^(2-3x) with limit  0 to 1 using limit of sums

identify a =0 , b=1

nh = b-a = 1

f(x)  = e^(2-3x)

find f(a) = f(0)
f(a+h)=f(h)
f(a+2h) = f[2h]
till the pattern can be identified,
f(a + (n-1) h ) = f[(n-1) h] etc

simplify using properties of sum of  n terms of a GP.
and use the limit [(e^h) -1] / h tends to 1 as h tends to 0

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

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

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

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

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

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