Advertisements
Advertisements
Question
Integrate the rational function:
`x/((x -1)^2 (x+ 2))`
Advertisements
Solution
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 ...(1)
Put x = 1
1 = 3B
⇒ B = `1/3`
Put x = -2
-2 = C (-2 - 1)2
⇒ C = `(-2)/9`
On comparing the coefficients of x2
A = `-C = 2/9`
Hence, `int x/((x+ 1)^2(x - 2))` dx
`= int 2/ (9 (x - 1)) dx + int 1/ (3 (x - 1)^2) dx - int 2/ (9(x + 2)) dx`
`= 2/9 log abs (x - 1) + 1/3 int (x - 1)^-1/-1 - 2/9 log abs (x + 2) + C`
`= 2/9 log abs ((x - 1)/(x + 2)) - 1/(3(x + 1)) + C`
APPEARS IN
RELATED QUESTIONS
Find : `int x^2/(x^4+x^2-2) dx`
Evaluate:
`int x^2/(x^4+x^2-2)dx`
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
Integrate the rational function:
`2/((1-x)(1+x^2))`
Integrate the rational function:
`(3x -1)/(x + 2)^2`
Integrate the rational function:
`(cos x)/((1-sinx)(2 - sin x))` [Hint: Put sin x = t]
Integrate the rational function:
`1/(e^x -1)`[Hint: Put ex = t]
Integrate the following w.r.t. x : `(x^2 + 2)/((x - 1)(x + 2)(x + 3)`
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(x^5 + 1)`
Integrate the following with respect to the respective variable : `(6x + 5)^(3/2)`
Integrate the following w.r.t.x : `(1)/(2cosx + 3sinx)`
Evaluate: `int (2"x" + 1)/(("x + 1")("x - 2"))` dx
`int (2x - 7)/sqrt(4x- 1) dx`
`int x^7/(1 + x^4)^2 "d"x`
`int ((x^2 + 2))/(x^2 + 1) "a"^(x + tan^(-1_x)) "d"x`
`int (sinx)/(sin3x) "d"x`
`int 1/(2 + cosx - sinx) "d"x`
`int sec^2x sqrt(tan^2x + tanx - 7) "d"x`
`int "e"^x ((1 + x^2))/(1 + x)^2 "d"x`
`int ("d"x)/(2 + 3tanx)`
`int xcos^3x "d"x`
`int ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1]) "d"x`
Choose the correct alternative:
`int sqrt(1 + x) "d"x` =
`int 1/x^3 [log x^x]^2 "d"x` = p(log x)3 + c Then p = ______
`int (3"e"^(2"t") + 5)/(4"e"^(2"t") - 5) "dt"`
If `int(sin2x)/(sin5x sin3x)dx = 1/3log|sin 3x| - 1/5log|f(x)| + c`, then f(x) = ______
Evaluate the following:
`int (x^2"d"x)/(x^4 - x^2 - 12)`
If `int "dx"/((x + 2)(x^2 + 1)) = "a"log|1 + x^2| + "b" tan^-1x + 1/5 log|x + 2| + "C"`, then ______.
Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)dx`
Find : `int (2x^2 + 3)/(x^2(x^2 + 9))dx; x ≠ 0`.
Evaluate:
`int 2/((1 - x)(1 + x^2))dx`
Evaluate:
`int(2x^3 - 1)/(x^4 + x)dx`
When is a rational function called improper?
Which partial-fraction decomposition is appropriate for \[\frac{\mathrm{p}x+\mathrm{q}}{(x-\mathrm{a})(x-\mathrm{b})}\]?
Which partial-fraction decomposition is used for \[\frac{\mathrm{p}x+\mathrm{q}}{(x-\mathrm{a})^{2}}\]?
