Advertisements
Advertisements
Question
Integrate the following with respect to x:
`sin^-1 ((2x)/(1 + x^2))`
Advertisements
Solution
I = `int sin^-1 ((2x)/(1 + x^2)) ""d"x`
Put x = tan θ
dx = sec2θ dθ
I = `int sin^-1 ((2tantheta)/(1 + tan^2theta)) sec^2theta "d"theta`
= `int sin^-1 (sin2theta) sec^2theta "d"theta`
= `int 2theta sec^2theta "d"theta`
= `2int (theta) (sec^2theta "d"theta)` ........[Rater Example 11.34]
I = `2[thetatantheta - int tan theta "d"theta]`
= `2theta tan theta - 2 log |sectheta| + "c"` ......`(sec theta = sqrt(1 + tan^2theta))`
`int sin^-1 ((2x)/(1 + x^2)) "d"x = 2x tan^-1x - 2log |sqrt(1 + x^2)| + "c"`
= `2[x tan^-1x - log |sqrt(1 + x^2)|] + "c"`
APPEARS IN
RELATED QUESTIONS
Evaluate : `int (1+logx)/(x(2+logx)(3+logx))dx`
Evaluate : `int_0^1 "x" . "tan"^-1 "x" "dx"`
Integrate the following functions with respect to x :
(2x – 5)(3x + 4x)
Integrate the following functions with respect to x :
`(sin4x)/sinx`
Integrate the following functions with respect to x :
sin2 5x
Integrate the following functions with respect to x :
`(3x - 9)/((x - 1)(x + 2)(x^2 + 1))`
Integrate the following with respect to x :
`x/sqrt(1 + x^2)`
Integrate the following with respect to x:
9xe3x
Integrate the following with respect to x:
x3 sin x
Integrate the following with respect to x:
`"e"^("a"x) cos"b"x`
Integrate the following with respect to x:
`"e"^x ((x - 1)/(2x^2))`
Integrate the following with respect to x:
`"e"^x sec x(1 + tan x)`
Integrate the following with respect to x:
`"e"^x ((2 + sin 2x)/(1 + cos 2x))`
Find the integrals of the following:
`1/(6x - 7 - x^2)`
Find the integrals of the following:
`1/sqrt(x^2 - 4x + 5)`
Integrate the following with respect to x:
`(2x - 3)/(x^2 + 4x - 12)`
Integrate the following with respect to x:
`(5x - 2)/(2 + 2x + x^2)`
Choose the correct alternative:
`int ("d"x)/("e"^x - 1)` is
