Advertisements
Advertisements
प्रश्न
Integrate the following with respect to x :
`sqrt(x)/(1 + sqrt(x))`
Advertisements
उत्तर
`int sqrt(x)/(1 + sqrt(x)) "d"x`
Put `1 + sqrt(x)` = u
`1 + x^(1/2)` = u
`(0 + 1/2 x^(1/2 - 1)) "d"x` = du
`1/2 x^(- 1/2) "d"x` = du
`1/(2 x^(1/2)) "d"x` = du
`1/sqrt(x) "d"x` = 2 du
`int sqrt(x)/(1 + sqrt(x)) "d"x = int (sqrt(x) * sqrt(x))/(sqrt(x) * (1 + sqrt(x)) "d"x`
`int sqrt(x)/(1 + sqrt(x)) "d"x = int x/((1 + sqrt(x))) * 1/sqrt(x) "d"x` .......(1)
`1 + sqrt(x)` = u
`sqrt(x)= "u" - 1`
x = `("u" - 1)^2`
(1) ⇒ `int sqrt(x)/(1 + sqrt(x)) "d"x = int ("u" - 1)^2/"u" 2"du"`
= `2 int ("u"^2 - 2"u" + 1)/"u" "du"`
= `2 int ("u"^2/"u" - (2"u")/"u" + 1/"u") "du"`
= `2 int ("u" - 2 + 1/"u") "du"`
= `2[int "u" "du" - int 2 "du" + int 1/"u" "du"]`
= `2 ["u"^2/"u" - 2"u" + log|"u"|] + "c"`
= `2[(1 + sqrt(x))^2/2 - 2(1 + sqrt(x)) + log|1 + sqrt(x)|] + "c"`
`int sqrt(x)/(1 + sqrt(x)) "d"x = (1 + sqrt(x))^2 - 4(1 + sqrt(x)) + 2log|1 + sqrt(x)| + "c"`
APPEARS IN
संबंधित प्रश्न
Evaluate : `int (1+logx)/(x(2+logx)(3+logx))dx`
Evaluate:`int(tansqrtx)/sqrtxdx`
Integrate the following functions with respect to x :
`(x^3 + 4x^2 - 3x + 2)/x^2`
Integrate the following functions with respect to x :
`(3 + 4cosx)/(sin^2x)`
Integrate the following functions with respect to x :
`"e"^(x log "a") "e"^x`
Integrate the following functions with respect to x :
`(8^(1 + x) + 4^(1 - x))/2^x`
Integrate the following functions with respect to x :
`(x + 1)/((x + 2)(x + 3))`
Integrate the following functions with respect to x :
`x^3/((x - 1)(x - 2))`
Integrate the following with respect to x :
`x^2/(1 + x^6)`
Integrate the following with respect to x :
`(sin sqrt(x))/sqrt(x)`
Integrate the following with respect to x :
`(sin^-1 x)/sqrt(1 - x^2)`
Integrate the following with respect to x :
sin5x cos3x
Integrate the following with respect to x:
`"e"^x (tan x + log sec x)`
Integrate the following with respect to x:
`"e"^x sec x(1 + tan x)`
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((x + 1)^2 - 4)`
Choose the correct alternative:
`int ("e"^x(x^2 tan^-1x + tan^-1x + 1))/(x^2 + 1) "d"x` is
Choose the correct alternative:
`int 1/(x sqrt(log x)^2 - 5) "d"x` is
Choose the correct alternative:
`int sin sqrt(x) "d"x` is
