Advertisements
Advertisements
Question
`int x/((x - 1)^2 (x + 2)) "d"x`
Advertisements
Solution
Let I = `int x/((x - 1)^2 (x + 2)) "d"x`
Let `x/((x - 1)^2 (x + 2)) = "A"/(x - 1) + "B"/(x - 1)^2 + "C"/((x + 2))`
∴ x = A(x – 1)(x + 2) + B(x + 2) + C(x – 1)2 ......(i)
Putting x = 1 in (i), we get
1 = A(0)(3) + B(3) + C(0)2
∴ 1 = 3B
∴ B = `1/3`
Putting x = 2 in (i), we get
– 2 = A(– 3)(0) + B(0) + C(9)
∴ – 2 = 9C
∴ C = `-2/9`
Putting x = – 1 in (i), we get
– 1 = A(– 2)(1) + B(1) + C(4)
∴ – 1 = `-2"A" + 1/3 - 8/9`
∴ – 1 = `-2"A" - 5/9`
∴ 2A = `-5/9 + 1 = 4/9`
∴ A = `2/9`
∴ `x/((x - 1)^2(x + 2)) = (2/9)/(x - 1) + (1/3)/(x - 1)^2 + ((-2/9))/(x + 2)`
∴ I = `int[(2/9)/(x - 1) + (1/3)/(x - 1)^2 + ((-2/9))/(x + 2)] "d"x`
= `2/9 int 1/(x - 1) "d"x + 1/3int(x - 1)^(-2) "d"x - 2/9 int 1/(x + 2) "d"x`
= `2/9 log|x - 1| + 1/3*((x - 1)^(-1))/(-1) - 2/9 log|x + 2| + "c"`
= `2/9 log|x - 1| - 2/9 log|x + 2| - 1/3 xx 1/((x - 1)) + "c"`
∴ I = `2/9 log|(x - 1)/(x + 2)| - 1/(3(x - 1)) + "c"`
RELATED QUESTIONS
Integrate the rational function:
`(3x -1)/(x + 2)^2`
Integrate the rational function:
`((x^2 +1)(x^2 + 2))/((x^2 + 3)(x^2+ 4))`
Integrate the following w.r.t. x : `x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3))`
Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`
Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`
Integrate the following w.r.t. x : `(12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)`
Integrate the following w.r.t. x : `(1)/(x(1 + 4x^3 + 3x^6)`
Integrate the following w.r.t. x : `(1)/(x^3 - 1)`
Integrate the following w.r.t. x : `(1)/(2sinx + sin2x)`
Integrate the following w.r.t. x : `(5*e^x)/((e^x + 1)(e^(2x) + 9)`
Integrate the following w.r.t.x : `(1)/(2cosx + 3sinx)`
Integrate the following w.r.t.x:
`x^2/((x - 1)(3x - 1)(3x - 2)`
Integrate the following w.r.t.x : `sqrt(tanx)/(sinx*cosx)`
For `int ("x - 1")/("x + 1")^3 "e"^"x" "dx" = "e"^"x"` f(x) + c, f(x) = (x + 1)2.
`int sqrt(4^x(4^x + 4)) "d"x`
`int sec^3x "d"x`
`int (x^2 + x -1)/(x^2 + x - 6) "d"x`
Choose the correct alternative:
`int ((x^3 + 3x^2 + 3x + 1))/(x + 1)^5 "d"x` =
Evaluate `int (2"e"^x + 5)/(2"e"^x + 1) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
Evaluate the following:
`int x^2/(1 - x^4) "d"x` put x2 = t
Evaluate the following:
`int (x^2"d"x)/(x^4 - x^2 - 12)`
Evaluate the following:
`int sqrt(tanx) "d"x` (Hint: Put tanx = t2)
`int 1/(x^2 + 1)^2 dx` = ______.
Evaluate:
`int x/((x + 2)(x - 1)^2)dx`
Evaluate:
`int (x + 7)/(x^2 + 4x + 7)dx`
Which partial form is appropriate for \[\frac{\mathrm{p}x^{2}+\mathrm{q}x+\mathrm{r}}{(x-\mathrm{a})^{2}(x-\mathrm{b})}\]?
For \[\frac{5x-5}{(x-2)(x-3)}=\frac{\mathrm{A}}{x-2}+\frac{\mathrm{B}}{x-3}\], which equation results after clearing denominators?
