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 solid generated by rotating the area bounded by x2+y2 =36 and the lines x = 0, x = 3 about X -axis.
Integrate the following functions with respect to x :
`[sqrt(x) + 1/sqrt(x)]^2`
Integrate the following functions with respect to x :
`(cos 2x)/(sin^2x cos^2x)`
Integrate the following functions with respect to x :
`(3 + 4cosx)/(sin^2x)`
Integrate the following functions with respect to x :
cos 3x cos 2x
Integrate the following functions with respect to x :
`(3x + 4) sqrt(3x + 7)`
Integrate the following with respect to x :
`x^2/(1 + x^6)`
Integrate the following with respect to x :
`cosx/(cos(x - "a"))`
Integrate the following with respect to x:
27x2e3x
Integrate the following with respect to x:
`(x sin^-1 x)/sqrt(1 - x^2)`
Integrate the following with respect to x:\
`logx/(1 + log)^2`
Find the integrals of the following:
`1/(25 - 4x^2)`
Integrate the following with respect to x:
`(2x + 1)/sqrt(9 + 4x - x^2)`
Integrate the following functions with respect to x:
`sqrt(x^2 + 2x + 10)`
Integrate the following functions with respect to x:
`sqrt(81 + (2x + 1)^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 ("e"^(6 log x) - "e"^(5logx))/("e"^(4logx) - "e"^(3logx)) "d"x` is
Choose the correct alternative:
`int 2^(3x + 5) "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
