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`
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 :
`[sqrt(x) + 1/sqrt(x)]^2`
Integrate the following functions with respect to x :
cos 3x cos 2x
Integrate the following with respect to x :
`x/sqrt(1 + x^2)`
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^-1 x)/sqrt(1 - x^2)`
Integrate the following with respect to x :
`alpha beta x^(alpha - 1) "e"^(- beta x^alpha)`
Integrate the following with respect to x:
25xe–5x
Integrate the following with respect to x:
`(x sin^-1 x)/sqrt(1 - x^2)`
Integrate the following with respect to x:
x5ex2
Integrate the following with respect to x:
`tan^-1 ((8x)/(1 - 16x^2))`
Integrate the following with respect to x:
`"e"^(- 3x) sin 2x`
Integrate the following with respect to x:
`"e"^(- 3x) cos x`
Integrate the following with respect to x:
`"e"^x (tan x + log sec x)`
Find the integrals of the following:
`1/(4 - x^2)`
Choose the correct alternative:
`int 2^(3x + 5) "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 "e"^(- 7x) sin 5x "d"x` is
Choose the correct alternative:
`int x^2 "e"^(x/2) "d"x` is
