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

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


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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

 

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


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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 

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


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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 










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


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solving a equation involving exponentials

solve sinh (x)=0.5
 (eˣ - e⁻ˣ)/2 =0.5                   hyperbolic functions
so that eˣ - e⁻ˣ =1

 now solve eˣ - e⁻ˣ = 1

multiply with eˣ

make it into a quadratic in eˣ

solve using quadratic fomula to find the value of eˣ

take natural logarithm to find x

 

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proving a trigonometric identity

proving a trigonometric identity

change everything in terms of sinx and cosx
simplify, then  use trigonometric identities

trigonometry identities formulae etc


 










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integral using trigonometric substitution

integral of (x^3)/ [sqrt[(x^2)+4] ]
 use the substitution x = 2tanθ
 dx = 2 ((secθ)^2) dθ
 simplify and then use another substituion
secθ = t
secθ tanθ dθ = dt
simplify and integrate then resubstitute
 change secθ in terms of tanθ
then again resubstitute
 remembering that x =2 tanθ or tanθ = (x/2)

 
 

 
 
 integration formulae


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