Advertisements
Advertisements
Question
Integrate the following with respect to x:
`"e"^("a"x) cos"b"x`
Advertisements
Solution
`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
RELATED QUESTIONS
Integrate the following functions with respect to x :
(2x – 5)(3x + 4x)
Integrate the following functions with respect to x :
`(cos2x - cos 2 alpha)/(cosx - cos alpha)`
Integrate the following functions with respect to x :
`x^3/((x - 1)(x - 2))`
Integrate the following with respect to x :
`(sin 2x)/("a"^2 + "b"^2 sin^2x)`
Integrate the following with respect to x :
`alpha beta x^(alpha - 1) "e"^(- beta x^alpha)`
Integrate the following with respect to x :
`tan x sqrt(sec x)`
Integrate the following with respect to x:
`sin^-1 ((2x)/(1 + x^2))`
Integrate the following with respect to x:
`"e"^(-x) cos 2x`
Integrate the following with respect to x:
`"e"^x (tan x + log sec x)`
Integrate the following with respect to x:
`"e"^(tan^-1 x) ((1 + x + x^2)/(1 + x^2))`
Integrate the following with respect to x:\
`logx/(1 + log)^2`
Find the integrals of the following:
`1/sqrt(xx^2 + 4x + 2)`
Find the integrals of the following:
`1/sqrt((2 + x)^2 - 1)`
Integrate the following with respect to x:
`(x + 2)/sqrt(x^2 - 1)`
Integrate the following functions with respect to x:
`sqrt(x^2 + 2x + 10)`
Integrate the following functions with respect to x:
`sqrt(9 - (2x + 5)^2`
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 sin^2x "d"x` is
Choose the correct alternative:
`int sin sqrt(x) "d"x` is
