Advertisements
Advertisements
प्रश्न
Integrate the following with respect to x:
`"e"^("a"x) cos"b"x`
Advertisements
उत्तर
`int "e"^("a"x) cos"b"x`
Let I = `int "e"^("a"x) cos "b"x "d"x`
u = cos bx
⇒ dc = `"e"^("a"x) "d"x`
du = – b sin b dx
⇒ = `"e"^("a"x)/"a"`
Applying Integration by parts,
I = `"e"^("a"x)/"a" cos "b"x - int "e"^("a"x)/"a" (- "b" sin "b"x) "d"x`
I = `"e"^("a"x)/"a" cos "b"x + "b"/"a" int"e"^("a"x) sin "b"x "d"x`
u = sin x bx
⇒ dv = `"e"^("a"x) "d"x`
du = b cos bx dx
⇒ = `"e"^("a"x)/"a"`
Again Applying Integration by parts,
I = `"e"^("a"x)/"a" cos "b" + "b"/"a" ["e"^("a"x)/"a" sin "b"x - "b"/"a" int "e"^("a"x) cos bx "d"x]`
I = `("e"^("a"x) cos "b"x)/"a" + "b"/"a" [("e"^("a"x) sin "b"x)/"a" - "b"/"a" "I"]`
I = `("e"^("a"x) cos "b"x)/"a" + ("be"^("a"x) sin "b"x)/"a" - "b"^2/"a"^2 "I"`
`"I" + "b"^2/""^2 "I" = ("ae"^("a"x) cos"b"x + "be"^("a"x) sin"b"x)/"a"^2 + "c"`
`"I"(("a"^2 + "b"^2)/"a"^2) = ("e"^("a"x) ["a"cos"b"x + "b" sin"b"x])/"a"^2 + "c"`
∴ `int "e"^("a"x) cos "b"x "d"x = "e"^("a"x)/("a"^2 + "b"^2) ["a" cos "b"x + "b" sin "b"x] + "c"``
APPEARS IN
संबंधित प्रश्न
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 :
`(sin4x)/sinx`
Integrate the following functions with respect to x :
sin2 5x
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 :
`(3x - 9)/((x - 1)(x + 2)(x^2 + 1))`
Integrate the following with respect to x :
`(sin sqrt(x))/sqrt(x)`
Integrate the following with respect to x:
9xe3x
Integrate the following with respect to x:
`"e"^(- 4x) sin 2x`
Integrate the following with respect to x:
`"e"^x sec x(1 + tan x)`
Find the integrals of the following:
`1/sqrt(x^2 - 4x + 5)`
Find the integrals of the following:
`1/sqrt(9 + 8x - x^2)`
Integrate the following functions with respect to x:
`sqrt(x^2 - 2x - 3)`
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"^x (1 + x))/(cos^2(x"e"^x)) "d"x` is
Choose the correct alternative:
`int "e"^(- 4x) cos "d"x` is
Choose the correct alternative:
`int sin sqrt(x) "d"x` is
