Advertisements
Advertisements
Question
Integrate the following with respect to x:
`(x + 2)/sqrt(x^2 - 1)`
Advertisements
Solution
Let x + 2 = `"A" "d"/("d"x) (x^2 - 1) + "B"`
x + 2 = A(2x) + B
2A = 1
⇒ A = `1/2`
B = 2
x + 2 = `1/2 (2x) + 2`
`int (x + 2)/sqrt(x^2 - 1) "d"x = int (1/2 (2x) + 2)/sqrt(x^2 - 1) "d"x`
= `1/2 int (2x)/sqrt(x^2 - 1) "d"x + 2 int ("d"x)/sqrt(x^2 - 1)`
Put `x^2 - 1` = t
2x dx = dt
= `1/2 int "dt"/"t" + 2 int ("d"x)/sqrt(x^2 - 1^2)`
= `1/2 "t"^(- 1/2) "dt" + 2 log |x + sqrt(x^2 - 1^2)| + "c"`
= `1/2 ("t"^(- 1/2 + 1))/(- 1/2 + 1) + 2 log |x + sqrt(x^2 - 1)| + "c"`
= `1/2 ("t"^(1/2))/(1/2) + 2 log |x + sqrt(x^2 - 1)| + "c"`
`int (x + 2)/sqrt(x^2 - 1) "d"x = sqrt(x^2 - 1) + 2 log |x + sqrt(x^2 - 1)| + "c"`
APPEARS IN
RELATED QUESTIONS
Find the volume of the solid generated by the complete revolution of the ellipse `"x"^2/36 + "y"^2/25 = 1` about Y-axis.
Integrate the following functions with respect to x :
`(x^3 + 4x^2 - 3x + 2)/x^2`
Integrate the following functions with respect to x :
`[sqrt(x) + 1/sqrt(x)]^2`
Integrate the following functions with respect to x :
`(sin^2x)/(1 + cosx)`
Integrate the following functions with respect to x :
cos 3x cos 2x
Integrate the following functions with respect to x :
`1/(sqrt(x + 3) - sqrt(x - 4))`
Integrate the following functions with respect to x :
`(3x - 9)/((x - 1)(x + 2)(x^2 + 1))`
Integrate the following with respect to x :
`cot x/(log(sin x))`
Integrate the following with respect to x :
`(sin^-1 x)/sqrt(1 - x^2)`
Integrate the following with respect to x:
25xe–5x
Integrate the following with respect to x:
x sec x tan x
Find the integrals of the following:
`1/(9x^2 - 4)`
Find the integrals of the following:
`1/sqrt(xx^2 + 4x + 2)`
Integrate the following with respect to x:
`(2x + 1)/sqrt(9 + 4x - x^2)`
Integrate the following functions with respect to x:
`sqrt(9 - (2x + 5)^2`
Choose the correct alternative:
`int ("e"^(6 log x) - "e"^(5logx))/("e"^(4logx) - "e"^(3logx)) "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 sin sqrt(x) "d"x` is
