Advertisements
Advertisements
प्रश्न
Integrate the rational function:
`((x^2 +1)(x^2 + 2))/((x^2 + 3)(x^2+ 4))`
Advertisements
उत्तर
`((x^2 + 1)(x^2 + 2))/((x^2 + 3)(x^2 + 4))` Taking x2 = y
`((y + 1)(y + 2))/((y + 3)(y + 4)) = (y^2 + 3y + 2)/(y^2 + 7y + 12)`
`= 1 - (4y + 10)/(y^2 + 7y + 12)`
`= 1 - (4y + 10)/((y + 3)(y + 4))`
Let `(4y + 10)/((y + 3)(y + 4)) = A/((y + 3)) + B/(y + 4)`
4y + 10 = A (y + 4) + B (y + 3)
Putting y = -4 - 6 = 0 - B
⇒ B = 6
Putting y = -3, -2 = A + 0
⇒ A = -2
`therefore ((x^2 + 1)(x^2 + 2))/((x^2 + 3)(x^2 + 4)) = 1 - [(-2)/(y + 3) + 6/(y + 4)]`
`= 1 + 2/(y + 3) + 6/(y + 4)`
`int ((x^2 + 1)(x^2 + 2))/((x^2 + 3)(x^2 + 4))` dx
`= int dx + 2 int 1/(x^2 sqrt(3^2)) + 6 int 1/(x^2 + 4)` dx
`= x + 2/sqrt 3 tan^-1 x/sqrt3 - 6/2 tan^-1 (x/2) + C`
`= x + 2/sqrt 3 tan^-1 x/sqrt3 - 3 tan^-1 x/2 + C`
APPEARS IN
संबंधित प्रश्न
Evaluate:
`int x^2/(x^4+x^2-2)dx`
Integrate the rational function:
`x/((x-1)(x- 2)(x - 3))`
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`(3x + 5)/(x^3 - x^2 - x + 1)`
Integrate the rational function:
`(x^3 + x + 1)/(x^2 -1)`
Integrate the rational function:
`(3x -1)/(x + 2)^2`
`int (xdx)/((x - 1)(x - 2))` equals:
`int (dx)/(x(x^2 + 1))` equals:
Find :
`∫ sin(x-a)/sin(x+a)dx`
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 : `2^x/(4^x - 3 * 2^x - 4`
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)`
Evaluate: `int (2"x" + 1)/(("x + 1")("x - 2"))` dx
Evaluate:
`int x/((x - 1)^2(x + 2)) dx`
State whether the following statement is True or False.
If `int (("x - 1") "dx")/(("x + 1")("x - 2"))` = A log |x + 1| + B log |x - 2| + c, then A + B = 1.
`int x^7/(1 + x^4)^2 "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`
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 (5(x^6 + 1))/(x^2 + 1) "d"x` = x5 – ______ x3 + 5x + c
If f'(x) = `1/x + x` and f(1) = `5/2`, then f(x) = log x + `x^2/2` + ______ + c
`int 1/x^3 [log x^x]^2 "d"x` = p(log x)3 + c Then p = ______
Evaluate the following:
`int (x^2"d"x)/(x^4 - x^2 - 12)`
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`
Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`
If f(x) = `int(3x - 1)x(x + 1)(18x^11 + 15x^10 - 10x^9)^(1/6)dx`, where f(0) = 0, is in the form of `((18x^α + 15x^β - 10x^γ)^δ)/θ`, then (3α + 4β + 5γ + 6δ + 7θ) is ______. (Where δ is a rational number in its simplest form)
`int 1/(x^2 + 1)^2 dx` = ______.
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 x^4/((x - 1)(x^2 + 1))dx`.
Value of ∫ `(x^2 + 1)/((x − 1)(x − 2))`dx is ______.
