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 : `int1/(x(3+logx))dx`
Evaluate : `∫_0^(pi/2) (sinx.cosx)/(1 + sin^4x)`.dx
Evaluate : `int_0^1 "x" . "tan"^-1 "x" "dx"`
Integrate the following functions with respect to x :
`(sin^2x)/(1 + cosx)`
Integrate the following functions with respect to x :
`1/((x - 1)(x + 2)^2`
Integrate the following with respect to x :
`x^2/(1 + x^6)`
Integrate the following with respect to x :
`("e"^x - "e"^-x)/("e"^x + "e"^-x)`
Integrate the following with respect to x :
`(sin 2x)/("a"^2 + "b"^2 sin^2x)`
Integrate the following with respect to x :
sin5x cos3x
Integrate the following with respect to x:
27x2e3x
Integrate the following with respect to x:\
`logx/(1 + log)^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 with respect to x:
`(5x - 2)/(2 + 2x + x^2)`
Integrate the following with respect to x:
`(x + 2)/sqrt(x^2 - 1)`
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 ("e"^x(x^2 tan^-1x + tan^-1x + 1))/(x^2 + 1) "d"x` is
Choose the correct alternative:
`int sqrt((1 - x)/(1 + x)) "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
