Advertisements
Advertisements
प्रश्न
Integrate the rational function:
`(3x + 5)/(x^3 - x^2 - x + 1)`
Advertisements
उत्तर
Let `(3x + 5)/(x^3 - x^2 - x + 1)`
`= (3x + 5)/(x^2(x - 1) - 1(x - 1))`
`= (3x + 5)/((x^2 - 1)(x - 1))`
`= (3x + 5)/((x + 1)(x - 1)^2)`
`(3x + 5)/((x + 1)(x - 1)^2) = A/(x + 1) = B/(x - 1) + C/((x - 1)^2)`
3x + 5 = A(x - 1)2 + B(x2 - 1) + C(x + 1) ... (i)
Put x = 1
8 = 0 + 0 + 2C
⇒ C = 4
Put x = -1
2 = A(-2)2 + 0 = 0
⇒ A = `-1/2`
On comparing the coefficients of x2
0 = A + B
⇒ A = -A `= 1/2`
Hence, `(3x + 5)/(x^3 - x^2 - x + 1)`
`= -1/(2(x + 1)) + 1/(2(x - 1)) + 4/((x - 1)^2)`
On integrating,
`int (3x + 5)/(x^3 - x^2 - x + 1)`
`= -1/2 int 1/(x + 1) dx + 1/2 int 1/(x - 1) dx + 4 int 1/((x - 1)^2) dx`
`= -1/2 log abs (x + 1) + 1/2 log (x - 1) + 4 (1/((x - 1))) + C`
`= 1/2 log abs ((x + 1)/(x - 1)) - 4/((x - 1)) + C`
APPEARS IN
संबंधित प्रश्न
Find : `int x^2/(x^4+x^2-2) dx`
Integrate the rational function:
`x/((x + 1)(x+ 2))`
Integrate the rational function:
`1/(x(x^n + 1))` [Hint: multiply numerator and denominator by xn − 1 and put xn = t]
Integrate the rational function:
`1/(e^x -1)`[Hint: Put ex = t]
`int (xdx)/((x - 1)(x - 2))` equals:
Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`
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 : `(12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)`
Integrate the following w.r.t. x : `(1)/(x^3 - 1)`
Integrate the following w.r.t. x : `(1)/(sin2x + cosx)`
Integrate the following w.r.t. x : `(2log x + 3)/(x(3 log x + 2)[(logx)^2 + 1]`
Integrate the following w.r.t.x : `(1)/(2cosx + 3sinx)`
Evaluate:
`int x/((x - 1)^2(x + 2)) dx`
Evaluate: `int 1/("x"("x"^5 + 1))` dx
`int "dx"/(("x" - 8)("x" + 7))`=
`int 1/(x(x^3 - 1)) "d"x`
`int ((x^2 + 2))/(x^2 + 1) "a"^(x + tan^(-1_x)) "d"x`
`int 1/(2 + cosx - sinx) "d"x`
`int sin(logx) "d"x`
`int (x^2 + x -1)/(x^2 + x - 6) "d"x`
`int ("d"x)/(2 + 3tanx)`
Choose the correct alternative:
`int (x + 2)/(2x^2 + 6x + 5) "d"x = "p"int (4x + 6)/(2x^2 + 6x + 5) "d"x + 1/2 int 1/(2x^2 + 6x + 5)"d"x`, then p = ?
`int 1/x^3 [log x^x]^2 "d"x` = p(log x)3 + c Then p = ______
Evaluate `int x^2"e"^(4x) "d"x`
If `int(sin2x)/(sin5x sin3x)dx = 1/3log|sin 3x| - 1/5log|f(x)| + c`, then f(x) = ______
If `intsqrt((x - 7)/(x - 9)) dx = Asqrt(x^2 - 16x + 63) + log|x - 8 + sqrt(x^2 - 16x + 63)| + c`, then A = ______
Verify the following using the concept of integration as an antiderivative
`int (x^3"d"x)/(x + 1) = x - x^2/2 + x^3/3 - log|x + 1| + "C"`
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)
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.
Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`
Let g : (0, ∞) `rightarrow` R be a differentiable function such that `int((x(cosx - sinx))/(e^x + 1) + (g(x)(e^x + 1 - xe^x))/(e^x + 1)^2)dx = (xg(x))/(e^x + 1) + c`, for all x > 0, where c is an arbitrary constant. Then ______.
If `int 1/((x^2 + 4)(x^2 + 9))dx = A tan^-1 x/2 + B tan^-1(x/3) + C`, then A – B = ______.
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 x^4/((x - 1)(x^2 + 1))dx`.
