Advertisements
Advertisements
Question
Integrate the following with respect to x:
`(3x + 1)/(2x^2 - 2x + 3)`
Advertisements
Solution
Let 3x + 1 = `"A" "d"/("d"x) (2x^2 - 2x + 3) + "B"`
3x + 1 = A(4x – 2) + B
3x + 1 = 4Ax – 2A + B
4A = 3
⇒ A = `3/4`
– 2A + B = 1
`- 2 xx 3/4 + "B"` = 1
`3/2 + "B"` = 1
B = `1 + 3/2 = 5/2`
B = `5/2`
3x + 1 = `3/4 (4x - 2) + 5/2`
`int (3x + 1)/(2x^2 - 2x + 3) "d"x = int (3/4 (4x - 2) + 5/2)/(2x^2 - 2x + 3) "d"x`
= `3/4 int (4x - 2)/(2x^2 - 2x + 3) + 5/2 int ("d"x)/(2x^2 - 2x + 3)`
Put `2x^2 - 2x + 3` = t
`(4x - 2) "d"x` = dt
= `3/4 int "dt"/"t" + 5/2 int ("d"x)/(2(x^2 - x + 3/2))`
= `3/4 log |"t"| + 5/4 int ("d"x)/((x - 1/2)^2 - (1/2)^2 + 3/2)`
= `3/4 log |2x^2 - 2x + 3| + 5/4 int ("d"x)/((x - 1/2)^2 + - 1/4 + 3/2)`
= `3/4 log |2x^2 - 2x + 3| + 5/4 int ("d"x)/((x - 1/2)^2 + (6 - 1)/4`
= `3/4 log |2x^2 - 2x + 3| + 5/4 int ("d"x)/((x - 1/2)^2 + 5/4)`
= `3/4 log |2x^2 - 2x + 3| + 5/4 int ("d"x)/((x - 1/2)^2 + (sqrt(5)/2)^2`
`int ("d"x)/(x^2 + "a"^2) = 1/"a" tan^-1 (x/"a") + "c"`
= `3/4 log |2x^2 - 2x + 3| + 5/4 xx tan^-1 ((x - 1/2)/(sqrt(5)/2)) + "c"`
= `3/4 log |2x^2 - 2x + 3| + 5/4 xx 1/(sqrt(5)/2) tan^-1 ((2x - 1)/sqrt(5)) + "c"`
= `3/4 log |2x^2 - 2x + 3| + 5/4 xx 2/sqrt(5) tan^-1 ((2x - 1)/sqrt(5)) + "c"`
= `3/4 log |2x^2 - 2x + 3| + sqrt(5)/2 tan^-1 ((2x - 1)/sqrt(5)) + "c"`
APPEARS IN
RELATED QUESTIONS
Evaluate : `∫_0^(pi/2) (sinx.cosx)/(1 + sin^4x)`.dx
Evaluate : `int _0^1 ("x" . ("sin"^-1 "x")^2)/sqrt (1 - "x"^2)` dx
Integrate the following functions with respect to x :
(2x – 5)(3x + 4x)
Integrate the following functions with respect to x :
`(cos 2x)/(sin^2x cos^2x)`
Integrate the following functions with respect to x :
`(3x - 9)/((x - 1)(x + 2)(x^2 + 1))`
Integrate the following functions with respect to x :
`x^3/((x - 1)(x - 2))`
Integrate the following with respect to x :
`("cosec" x)/(log(tan x/2))`
Integrate the following with respect to x :
`sqrt(x)/(1 + sqrt(x))`
Integrate the following with respect to x :
x(1 – x)17
Integrate the following with respect to x:
`sin^-1 ((2x)/(1 + x^2))`
Integrate the following with respect to x:
`"e"^(-x) cos 2x`
Integrate the following with respect to x:
`"e"^(tan^-1 x) ((1 + x + x^2)/(1 + x^2))`
Find the integrals of the following:
`1/(25 - 4x^2)`
Find the integrals of the following:
`1/sqrt((2 + x)^2 - 1)`
Choose the correct alternative:
The gradient (slope) of a curve at any point (x, y) is `(x^2 - 4)/x^2`. If the curve passes through the point (2, 7), then the equation of the curve 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
