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 : `int _0^1 ("x" . ("sin"^-1 "x")^2)/sqrt (1 - "x"^2)` dx
Integrate the following functions with respect to x :
`(x + 1)/((x + 2)(x + 3))`
Integrate the following with respect to x :
`("e"^x - "e"^-x)/("e"^x + "e"^-x)`
Integrate the following with respect to x :
`(sin sqrt(x))/sqrt(x)`
Integrate the following with respect to x :
`(sin 2x)/("a"^2 + "b"^2 sin^2x)`
Integrate the following with respect to x :
`tan x sqrt(sec x)`
Integrate the following with respect to x :
sin5x cos3x
Integrate the following with respect to x:
x sec x tan x
Integrate the following with respect to x:
x log x
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)`
Integrate the following with respect to x:\
`logx/(1 + log)^2`
Find the integrals of the following:
`1/(25 - 4x^2)`
Find the integrals of the following:
`1/(6x - 7 - x^2)`
Find the integrals of the following:
`1/sqrt(9 + 8x - x^2)`
Integrate the following with respect to x:
`(2x + 3)/sqrt(x^2 + 4x + 1)`
Integrate the following functions with respect to x:
`sqrt(81 + (2x + 1)^2`
Choose the correct alternative:
`int secx/sqrt(cos2x) "d"x` is
Choose the correct alternative:
`int (sec^2x)/(tan^2 x - 1) "d"x`
Choose the correct alternative:
`int "e"^(sqrt(x)) "d"x` is
