Advertisements
Advertisements
प्रश्न
`int x/((x - 1)^2 (x + 2)) "d"x`
Advertisements
उत्तर
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"`
संबंधित प्रश्न
Integrate the rational function:
`1/(x^2 - 9)`
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
Integrate the rational function:
`x/((x^2+1)(x - 1))`
Integrate the rational function:
`1/(x^4 - 1)`
Integrate the rational function:
`((x^2 +1)(x^2 + 2))/((x^2 + 3)(x^2+ 4))`
`int (xdx)/((x - 1)(x - 2))` equals:
Integrate the following w.r.t. x:
`(6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)`
Integrate the following w.r.t. x : `(2x)/((2 + x^2)(3 + x^2)`
Integrate the following w.r.t. x : `(5x^2 + 20x + 6)/(x^3 + 2x ^2 + x)`
Integrate the following w.r.t. x : `(1)/(x^3 - 1)`
Integrate the following w.r.t. x : `(5*e^x)/((e^x + 1)(e^(2x) + 9)`
Integrate the following w.r.t.x : `x^2/sqrt(1 - x^6)`
Evaluate:
`int x/((x - 1)^2(x + 2)) dx`
Evaluate: `int 1/("x"("x"^"n" + 1))` dx
`int (sinx)/(sin3x) "d"x`
`int x sin2x cos5x "d"x`
`int xcos^3x "d"x`
`int (sin2x)/(3sin^4x - 4sin^2x + 1) "d"x`
`int ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1]) "d"x`
The numerator of a fraction is 4 less than its denominator. If the numerator is decreased by 2 and the denominator is increased by 1, the denominator becomes eight times the numerator. Find the fraction.
If `intsqrt((x - 5)/(x - 7))dx = Asqrt(x^2 - 12x + 35) + log|x| - 6 + sqrt(x^2 - 12x + 35) + C|`, then A = ______.
Find : `int (2x^2 + 3)/(x^2(x^2 + 9))dx; x ≠ 0`.
Evaluate`int(5x^2-6x+3)/(2x-3)dx`
Which pair of equations is obtained by equating coefficients in \[5x-5=\mathrm{A}(x-3)+\mathrm{B}(x-2)\]?
What must be included for a repeated linear factor?
What numerator is used for an irreducible quadratic factor?
