Advertisements
Advertisements
प्रश्न
Integrate the following with respect to x:
`"e"^(tan^-1 x) ((1 + x + x^2)/(1 + x^2))`
Advertisements
उत्तर
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
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.
Evaluate : `int_0^1 "x" . "tan"^-1 "x" "dx"`
Integrate the following functions with respect to x :
`(cos 2x)/(sin^2x cos^2x)`
Integrate the following functions with respect to x :
cos 3x cos 2x
Integrate the following functions with respect to x :
`(8^(1 + x) + 4^(1 - x))/2^x`
Integrate the following with respect to x :
`x^2/(1 + x^6)`
Integrate the following with respect to x :
`(sin 2x)/("a"^2 + "b"^2 sin^2x)`
Integrate the following with respect to x:
x sin 3x
Integrate the following with respect to x:
`tan^-1 ((8x)/(1 - 16x^2))`
Integrate the following with respect to x:
`"e"^(-x) cos 2x`
Find the integrals of the following:
`1/((x + 1)^2 - 25)`
Integrate the following functions with respect to x:
`sqrt(x^2 - 2x - 3)`
Integrate the following functions with respect to x:
`sqrt((x + 1)^2 - 4)`
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 sin^2x "d"x` is
Choose the correct alternative:
`int secx/sqrt(cos2x) "d"x` is
Choose the correct alternative:
`int ("e"^x(x^2 tan^-1x + tan^-1x + 1))/(x^2 + 1) "d"x` is
Choose the correct alternative:
`int 1/(x sqrt(log x)^2 - 5) "d"x` is
Choose the correct alternative:
`int "e"^(sqrt(x)) "d"x` is
