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 (1+logx)/(x(2+logx)(3+logx))dx`
Integrate the following functions with respect to x :
`1/(sqrt(x + 3) - sqrt(x - 4))`
Integrate the following with respect to x :
`(10x^9 + 10^x log_"e" 10)/(10^x + x^10)`
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 :
`1/(x log x log (log x))`
Integrate the following with respect to x:
x sin 3x
Integrate the following with respect to x:
`(x sin^-1 x)/sqrt(1 - x^2)`
Integrate the following with respect to x:
`sin^-1 ((2x)/(1 + x^2))`
Integrate the following with respect to x:
`"e"^("a"x) cos"b"x`
Find the integrals of the following:
`1/(9x^2 - 4)`
Find the integrals of the following:
`1/((x + 1)^2 - 25)`
Find the integrals of the following:
`1/sqrt(x^2 - 4x + 5)`
Integrate the following functions with respect to x:
`sqrt(x^2 + 2x + 10)`
Integrate the following functions with respect to x:
`sqrt(x^2 - 2x - 3)`
Integrate the following functions with respect to x:
`sqrt(81 + (2x + 1)^2`
Choose the correct alternative:
If `int 3^(1/x)/x^2 "d"x = "k"(3^(1/x)) + "c"`, then the value of k is
Choose the correct alternative:
`int sqrt((1 - x)/(1 + x)) "d"x` is
Choose the correct alternative:
`int 1/(x sqrt(log x)^2 - 5) "d"x` is
