mixture: cbse 12th integration

Showing posts with label cbse 12th integration. Show all posts

Tuesday, January 24, 2017

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

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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.

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

Monday, January 16, 2017

integral using partial fractions

integral using partial fractions

integral ∫ dx / [ (x-2) (1 + x²)]

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 ∫ { cosx/ [ sinx +cosx] } dx

integral ∫ { cosx / [ sinx +cosx] } dx

first express the numerator in terms of the denominator and its derivative
split into two integrals ;one of them is of the form u' / u which will integrate to ln(u)

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

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

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