Advertisements
Advertisements
Question
Find the integrals of the following:
`1/sqrt(x^2 - 4x + 5)`
Advertisements
Solution
`int ("d"x)/sqrt(x^2 - 4x + 5) = int ("d"x)/sqrt((x - 2)^2 - 2^2 + 5)`
= `int ("d"x)/sqrt((x - 2)^2 - 4 + 5)`
= `int ("d"x)/sqrt((x - 2)^2 + 1)`
Put x – 2 = t
dx = dt
`int ("d"x)/sqrt(x^2 - 4x + 5) = int ("d"x)/sqrt("t"^2 + 1^2)`
`int ("d"x)/sqrt(x^2 + "a"^2) = log |x + sqrt(x^2 + "a"^2)| + "c"`
= `log|"t" + sqrt("t"^2 + 1^2)| + "c"`
= `log|(x - 2) + sqrt((x - 2)^2 + 1)| + "c"`
= `log|(x - 2) + sqrt(x^2 - 4x + 4x +1)| + "c"`
`int ("d"x)/sqrt(x^2 - 4x + 5) = log |(x - 2) + sqrt(x^2 - 4x + 5)| + "c"`
APPEARS IN
RELATED QUESTIONS
Evaluate : `int1/(x(3+logx))dx`
Evaluate : `int _0^1 ("x" . ("sin"^-1 "x")^2)/sqrt (1 - "x"^2)` dx
Integrate the following functions with respect to x :
`(3 + 4cosx)/(sin^2x)`
Integrate the following functions with respect to x :
`(sin^2x)/(1 + cosx)`
Integrate the following functions with respect to x :
`(3x - 9)/((x - 1)(x + 2)(x^2 + 1))`
Integrate the following with respect to x :
`sqrt(x)/(1 + sqrt(x))`
Integrate the following with respect to x :
`cosx/(cos(x - "a"))`
Integrate the following with respect to x:
x sec x tan x
Integrate the following with respect to x:
`"e"^(2x) sinx`
Integrate the following with respect to x:
`"e"^(-x) cos 2x`
Integrate the following with respect to x:\
`logx/(1 + log)^2`
Integrate the following with respect to x:
`(2x + 1)/sqrt(9 + 4x - x^2)`
Integrate the following with respect to x:
`(2x + 3)/sqrt(x^2 + 4x + 1)`
Integrate the following functions with respect to x:
`sqrt((x + 1)^2 - 4)`
Choose the correct alternative:
If `int 3^(1/x)/x^2 "d"x = "k"(3^(1/x)) + "c"`, then the value of k is
Choose the correct alternative:
The gradient (slope) of a curve at any point (x, y) is `(x^2 - 4)/x^2`. If the curve passes through the point (2, 7), then the equation of the curve is
Choose the correct alternative:
`int sqrt(tanx)/(sin2x) "d"x` is
Choose the correct alternative:
`int "e"^(- 4x) cos "d"x` is
Choose the correct alternative:
`int "e"^(- 7x) sin 5x "d"x` is
Choose the correct alternative:
`int (x + 2)/sqrt(x^2 - 1) "d"x` is
