mixture: integral calculus

Showing posts with label integral calculus. Show all posts

Monday, March 11, 2013

integral using manipulation of numerator in terms of denominator

integral of 1/[sin²x cos²x]

since sin²x + cos²x =1
rewrite the numerator as sin²x + cos²x
divide each term with the denominator
simplify and then integrate

study integration formulae

free guide on integration or integral calculus for cbse isc kerala higher secondary class 12 ncert scert students

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integral using substitution

integral of [√(tanx)] / [sinx cosx]

first introduce [√(tanx)] in the numerator and denominator

then simplify and change the tanx in the numerator in terms of sinx and cosx
simplify in terms of secx and tanx

then use substitution method

study integration formulae

Saturday, March 9, 2013

integral using trigonometric manipulation

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

we will try to write the numerator in terms of the denominator

so introduce sin(a-b) in the numerator and denominator

note that (x-b)-(x-a) =a-b

so write sin(a-b) as sin[ (x-b)-(x-a) ] and expand using sin[A-B] formula

divide the numerator term by term with the denominator

reduce to terms involving sec and then integrate

study integration formulae a

integral 5

integral of 1 / [x - √x ]

we have to use substitution method

take √x common in the denominator

put √x =t
study differentiation formulae ands integration formulae
then take the differential

(1/[2√x] ) dx = dt

rearrange the integral and rewrite it in terms of t in a single step
and then integrate

note that log x stands for natural logarithm ln x in this question

integral 4

integral of (log x )² / x
note that log x stands for natural logarithm ln x in this question

since there is no quotient rule in integration and since this is not a standard form you have to try to use substitution method to get to a standard form

so you put log x =t then take the differential

so (1/x) dx = dt

after that arrange the integral so that you can rewrite the integral in terms of t in a single step

then you get a standard integral which can be integrated.
study integration formulae

integral 3

integral of sec²x/cosec²x

since there is no quotient rule in integration, change everything in terms of sinx and cosx simplify to get tan²x
since there is no direct integral for tan²x change to sec²x-1 using trigonometry formulae and then integrate term by term.
study integration formulae
and

trigonometry identities formulae etc

integral 2

integral of [ x^3 - x^2 + x -1] / (x-1)

factorise the numerator

cancel off the common factor

use the integral of x^n to integrate term by term
go through the integration formulae

Friday, March 8, 2013

integral1

integral of (x^3 +3x +4) /sqrt(x)

since quotient rule is not used in integration you divide each term in the numerator with the denominator and
reduce every thing to x^n form and then integrate term by term
go through the integration formulae

mixture

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