Advertisements
Advertisements
Question
Integrate the following w.r.t.x:
`x^2/((x - 1)(3x - 1)(3x - 2)`
Advertisements
Solution
Let I = `int x^2/((x - 1)(3x - 1)(3x - 2)).dx`
Let `x^2/((x - 1)(3x - 1)(3x - 2)) = "A"/(x - 1) + "B"/(3x -1) + "C"/(3x - 2)`
∴ x2 = A(3x -1)(3x - 2) + B(x – 1)(3x - 2) + C(x – 1)(3x -1)
Put x – 1 = 0, i.e. x = 1, we get
∴ x2 = A(2)(1) + B(0)(1) + C(0)(2)
∴ 2 = 4A
∴ A = `(1)/(2)`
Put x + 2 = 0, i.e. x = – 2, we get
2 + 2 = A(0)(1) + B(– 3)(1) + C(– 3)(0)
∴ 6 = – 3B
∴ B = – 2
Put x + 3 = 0, i.e. x = – 3we get
9 + 2 = A(– 1)(0) + B(– 4)(0) + C(– 4)(– 1)
∴ 11 = 4C
∴ C = `(11)/(4)`
∴ `(x^2 + 2)/((3x - 1)(x - 1)(3x - 2))`
= `((1/4))/(3x - 1) + (-2)/(x - 1) + ((11/4))/(3x - 2)`
∴ I = `int [((1/4))/(3x - 1) + (-2)/(x - 1) + ((11/4))/(3x - 2)].dx`
= `(1)/(18) int (1)/(3x - 1).dx - 2 int(1)/(x - 1).dx + (4)/(9) int (1)/(3x - 2).dx`
= `(1)/(18) log|3x - 1| + (1)/(2) log|x - 1| - (4)/(9) log|3x - 2| + 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`
Evaluate: `∫8/((x+2)(x^2+4))dx`
Integrate the rational function:
`x/((x-1)(x- 2)(x - 3))`
Integrate the rational function:
`(5x)/((x + 1)(x^2 - 4))`
Integrate the rational function:
`(x^3 + x + 1)/(x^2 -1)`
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 following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`
Integrate the following w.r.t. x:
`(6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)`
Integrate the following w.r.t. x : `(12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)`
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 : `((3sin - 2)*cosx)/(5 - 4sin x - cos^2x)`
Integrate the following w.r.t. x : `(5*e^x)/((e^x + 1)(e^(2x) + 9)`
Integrate the following with respect to the respective variable : `cot^-1 ((1 + sinx)/cosx)`
Integrate the following w.r.t.x : `x^2/sqrt(1 - x^6)`
Integrate the following w.r.t.x : `(1)/((1 - cos4x)(3 - cot2x)`
Integrate the following w.r.t.x: `(x + 5)/(x^3 + 3x^2 - x - 3)`
Evaluate:
`int (2x + 1)/(x(x - 1)(x - 4)) dx`.
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` dx
Evaluate: `int 1/("x"("x"^"n" + 1))` dx
`int (2x - 7)/sqrt(4x- 1) dx`
`int x^2sqrt("a"^2 - x^6) "d"x`
`int ((x^2 + 2))/(x^2 + 1) "a"^(x + tan^(-1_x)) "d"x`
`int sqrt((9 + x)/(9 - x)) "d"x`
`int sec^3x "d"x`
`int sin(logx) "d"x`
`int (6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1) "d"x`
Evaluate:
`int (5e^x)/((e^x + 1)(e^(2x) + 9)) dx`
`int (sin2x)/(3sin^4x - 4sin^2x + 1) "d"x`
`int ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1]) "d"x`
`int (5(x^6 + 1))/(x^2 + 1) "d"x` = x5 – ______ x3 + 5x + c
`int 1/x^3 [log x^x]^2 "d"x` = p(log x)3 + c Then p = ______
State whether the following statement is True or False:
For `int (x - 1)/(x + 1)^3 "e"^x"d"x` = ex f(x) + c, f(x) = (x + 1)2
`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^2 + "a"^2)(x^2 + "b"^2))`
Evaluate the following:
`int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x`
Evaluate the following:
`int sqrt(tanx) "d"x` (Hint: Put tanx = t2)
Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)dx`
If `int dx/sqrt(16 - 9x^2)` = A sin–1 (Bx) + C then A + B = ______.
If `int 1/((x^2 + 4)(x^2 + 9))dx = A tan^-1 x/2 + B tan^-1(x/3) + C`, then A – B = ______.
Evaluate: `int (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`
Find : `int (2x^2 + 3)/(x^2(x^2 + 9))dx; x ≠ 0`.
Evaluate:
`int x/((x + 2)(x - 1)^2)dx`
