Advertisements
Advertisements
Question
Integrate the following with respect to x:
`"e"^(tan^-1 x) ((1 + x + x^2)/(1 + x^2))`
Advertisements
Solution
I = `int "e"^(tan^-1 x) ((1 + x + x^2)/(1 + x^2)) "d"x`
I = `int "e"^(tan^-1 x) (1 + x + x^2) 1/(1 + x^2) "d"x`
Put tan–1 x = t
`1/(x1 + x^2) "d"x` = dt
x = tan t
∴ I = `int "e"^"t" [1 + tan "t" + tan^2"t"] "dt"`
= `int "e"^"t" [1 + tan^2"t" + tan "t"] "dt"`
= `int "e"^"t" [sec^2"t" + tan "t"] "dt"`
= `int "e"^"t" [tan "t" + sec^2"t"] "dt"`
f(x) = tan t
f'(x) = sec2t
`[int "e"^x ["f"(x) + "f"(x)] "d"x = "e"^x "f"(x) + "c"]`
∴ I = et tan t + c
I = `"e"^(tan^-1 (x)) * x + "c"`
I = `"e"^(tan^-1(x)) + "c"`
APPEARS IN
RELATED QUESTIONS
Evaluate : `int1/(x(3+logx))dx`
Evaluate : `int (1+logx)/(x(2+logx)(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.
Integrate the following functions with respect to x :
`(cos2x - cos 2 alpha)/(cosx - cos alpha)`
Integrate the following functions with respect to x :
`(sin^2x)/(1 + cosx)`
Integrate the following functions with respect to x :
`(x + 1)/((x + 2)(x + 3))`
Integrate the following functions with respect to x :
`1/((x - 1)(x + 2)^2`
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:
x3 sin x
Integrate the following with respect to x:
`(x sin^-1 x)/sqrt(1 - x^2)`
Integrate the following with respect to x:
`"e"^(- 3x) sin 2x`
Integrate the following with respect to x:
`"e"^x sec x(1 + tan x)`
Integrate the following with respect to x:
`(x + 2)/sqrt(x^2 - 1)`
Choose the correct alternative:
`int sqrt(tanx)/(sin2x) "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 ("d"x)/("e"^x - 1)` is
Choose the correct alternative:
`int x^2 "e"^(x/2) "d"x` is
Choose the correct alternative:
`int 1/(x sqrt(log x)^2 - 5) "d"x` is
