Advertisements
Advertisements
Question
Integrate the following with respect to x:\
`logx/(1 + log)^2`
Advertisements
Solution
I = `int logx/(1 + log x)^2 "d"x`
Put t = ` log x`
⇒ x = et
dx = et dt
I = `int "t"/(1 + "t")^2 "e"^"t" * "dt"`
I = `int "e"^"t" (1 + "t" - 1)/(1 + "t")^2 "dt"`
I = `int "e"^"t" [(1 + "t")/(1 + "t")^2 - 1/(1 + "t")^2] "dt"`
I = `int "e"^"t" [1/(1 + "t") - 1/(1 + "t")^2] "dt"`
f(t) = `1/(1 + "t")`
f'(t) = `- 1/(1 + "t")^2`
I = `"e"^"t" * 1/(1 + "t") + "c"`
= `x * 1/(1 + log x) + "c"`
I = `x/(1 + log x) + "c"`
APPEARS IN
RELATED QUESTIONS
Evaluate : `∫_0^(pi/2) (sinx.cosx)/(1 + sin^4x)`.dx
Evaluate : `int_0^1 "x" . "tan"^-1 "x" "dx"`
Evaluate : `int _0^1 ("x" . ("sin"^-1 "x")^2)/sqrt (1 - "x"^2)` dx
Integrate the following functions with respect to x :
`(1 + cos 4x)/(cos x - tan x)`
Integrate the following functions with respect to x :
`(3x + 4) sqrt(3x + 7)`
Integrate the following functions with respect to x :
`1/(sqrt(x + 3) - sqrt(x - 4))`
Integrate the following with respect to x :
`x/sqrt(1 + x^2)`
Integrate the following with respect to x :
`(10x^9 + 10^x log_"e" 10)/(10^x + x^10)`
Integrate the following with respect to x:
x sec x tan x
Integrate the following with respect to x:
27x2e3x
Integrate the following with respect to x:
x3 sin x
Integrate the following with respect to x:
`"e"^(-x) cos 2x`
Find the integrals of the following:
`1/(4 - x^2)`
Find the integrals of the following:
`1/(6x - 7 - x^2)`
Find the integrals of the following:
`1/sqrt(x^2 - 4x + 5)`
Integrate the following functions with respect to x:
`sqrt((6 - x)(x - 4))`
Integrate the following functions with respect to x:
`sqrt(9 - (2x + 5)^2`
Choose the correct alternative:
`int "e"^(- 4x) cos "d"x` is
Choose the correct alternative:
`int "e"^(sqrt(x)) "d"x` is
