Advertisements
Advertisements
प्रश्न
Integrate the following with respect to x:
27x2e3x
उत्तर
`int 27x^2"e"^(3x) "d"x = 27 int x^2 "e"^(3x) "d"x`
u = x2
u'= 2x
u" = 2
u"' = 0
v = `"e"^(3x) * "d"x`
⇒ v = `int "e"^(3x) * "d"x`
= `("e"^(3x))/3`
v1 = `int "v" "d"x`
= `int ("e"^(3x))/3 "d"x`
= `1/3 xx ("e"^3x)/3`
= `("e"^(3x))/3^2`
v2 = `int "v"_1 "d"x`
= `int ("e"^(3x))/3^2 "d"x`
= `1/3^2 xx ("e"^3x)/3`
= `1/3^3 "e"^(3x)`
v3 = `int "v"_2 "d"x`
= `int 1/3^3 "e"^(3x) "d"x`
= `1/3^4 "e"^(3x)`
`int "u" "dv"` = uv – u'v1 + u"v2 – u"'v3 + ..........
`27 int x^2 "e"^(3x) "d"x = 27[x^2 "e"^(3x)/3 - 2x "e"^(3x)/3^2 + 2 xx "e"^(3x)/3^3 - 0 xx "e"^(3x)/3^4] + "c"`
= `9x^2 "e"^(3x) - 6x "e"^(3x) + 2"e"^(3x) + 0 + "c"`
`27int x^2 "e"^(3x) "d"x = (9x^2 - 6x + 2)"e"^(3x) + "c"`
APPEARS IN
संबंधित प्रश्न
Evaluate : `int (1+logx)/(x(2+logx)(3+logx))dx`
Find the volume of the solid obtained by revolving about the X-axis, the region bounded by the curve `"x"^2/4 - "y"^2/9 = 1` and the lines x = 2 , x = 4.
Integrate the following functions with respect to x :
(2x – 5)(3x + 4x)
Integrate the following functions with respect to x :
`(sin^2x)/(1 + cosx)`
Integrate the following functions with respect to x :
`(1 + cos 4x)/(cos x - tan x)`
Integrate the following functions with respect to x :
`(x + 1)/((x + 2)(x + 3))`
Integrate the following with respect to x :
`cot x/(log(sin x))`
Integrate the following with respect to x :
`(sin^-1 x)/sqrt(1 - x^2)`
Integrate the following with respect to x:
x3 sin x
Integrate the following with respect to x:
x5ex2
Integrate the following with respect to x:
`"e"^x (tan x + log sec x)`
Find the integrals of the following:
`1/sqrt((2 + x)^2 - 1)`
Integrate the following functions with respect to x:
`sqrt(81 + (2x + 1)^2`
Integrate the following functions with respect to x:
`sqrt((x + 1)^2 - 4)`
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 (1 + x))/(cos^2(x"e"^x)) "d"x` 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 "e"^(- 7x) sin 5x "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
