Advertisements
Advertisements
Question
Integrate the following with respect to x:
x3 sin x
Solution
`int x^3 sin x`
u = x3
u’ = 3x2
u” = 6x
u”’ = 6
u’v = 0
dv = sin x dx
⇒ v = `int sin x "d"x`
= – cosx
v1 = `int "v" "d"x`
= `int - cos x "d"x`
= `- int cos x "d"x`
= – sin x
v2 = `int "v"_1 "d"x`
= `int - sin x "d"x`
= `- int sin x "d"x`
= – (– cos x)
= cos x
v3 = `int "v"_2 "d"x`
= `int cos x "d"x`
= sin x
v4 = `int "v"_3 "d"x`
= `int sin x "d"x`
= – cos x
`int "u" "dv"` = uv – u’ v1 + u” v2 – u”’ v3 + u’vv4 – ………..
`int x^3 sin x "d"x = x^3 (- cos x) – 3x^2 (- sin x) + 6x (cos x) – 6 sin x + 0 (- cos x) + "c"`
= – x3 cos x + 3x2 sin x + 6x cos x – 6 sin x + c
APPEARS IN
RELATED QUESTIONS
Evaluate : `int1/(x(3+logx))dx`
Find the volume of solid generated by rotating the area bounded by x2+y2 =36 and the lines x = 0, x = 3 about X -axis.
Evaluate : `∫_0^(pi/2) (sinx.cosx)/(1 + sin^4x)`.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.
Evaluate : `int _0^1 ("x" . ("sin"^-1 "x")^2)/sqrt (1 - "x"^2)` dx
Integrate the following functions with respect to x :
`(3x + 4) sqrt(3x + 7)`
Integrate the following with respect to x :
`("e"^x - "e"^-x)/("e"^x + "e"^-x)`
Integrate the following with respect to x :
`(sin^-1 x)/sqrt(1 - x^2)`
Integrate the following with respect to x :
`alpha beta x^(alpha - 1) "e"^(- beta x^alpha)`
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:
`sin^-1 ((2x)/(1 + x^2))`
Integrate the following with respect to x:
`"e"^(- 3x) sin 2x`
Integrate the following with respect to x:
`(3x + 1)/(2x^2 - 2x + 3)`
Integrate the following with respect to x:
`(2x + 3)/sqrt(x^2 + 4x + 1)`
Choose the correct alternative:
`int ("e"^x (1 + x))/(cos^2(x"e"^x)) "d"x` is
Choose the correct alternative:
`int x^2 cos x "d"x` is
Choose the correct alternative:
`int sqrt((1 - x)/(1 + x)) "d"x` is
Choose the correct alternative:
`int "e"^(- 7x) sin 5x "d"x` is
Choose the correct alternative:
`int sin sqrt(x) "d"x` is
