Advertisements
Advertisements
Question
Integrate the rational function:
`((x^2 +1)(x^2 + 2))/((x^2 + 3)(x^2+ 4))`
Advertisements
Solution
`((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
RELATED QUESTIONS
Integrate the rational function:
`(5x)/((x + 1)(x^2 - 4))`
Integrate the rational function:
`(x^3 + x + 1)/(x^2 -1)`
Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`
Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`
Integrate the following w.r.t. x : `(1)/(x(x^5 + 1)`
Integrate the following w.r.t. x : `2^x/(4^x - 3 * 2^x - 4`
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(1 + 4x^3 + 3x^6)`
Choose the correct options from the given alternatives :
If `int tan^3x*sec^3x*dx = (1/m)sec^mx - (1/n)sec^n x + c, "then" (m, n)` =
Integrate the following w.r.t. x: `(x^2 + 3)/((x^2 - 1)(x^2 - 2)`
Integrate the following with respect to the respective variable : `(cos 7x - cos8x)/(1 + 2 cos 5x)`
Integrate the following w.r.t.x:
`x^2/((x - 1)(3x - 1)(3x - 2)`
Integrate the following w.r.t.x : `sec^2x sqrt(7 + 2 tan x - tan^2 x)`
Evaluate:
`int (2x + 1)/(x(x - 1)(x - 4)) dx`.
Evaluate:
`int x/((x - 1)^2(x + 2)) dx`
Evaluate: `int 1/("x"("x"^"n" + 1))` dx
Evaluate: `int (5"x"^2 + 20"x" + 6)/("x"^3 + 2"x"^2 + "x")` dx
`int "dx"/(("x" - 8)("x" + 7))`=
`int (7 + 4x + 5x^2)/(2x + 3)^(3/2) dx`
`int (sinx)/(sin3x) "d"x`
`int "e"^x ((1 + x^2))/(1 + x)^2 "d"x`
`int (x^2 + x -1)/(x^2 + x - 6) "d"x`
`int xcos^3x "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 = ?
If f'(x) = `1/x + x` and f(1) = `5/2`, then f(x) = log x + `x^2/2` + ______ + c
`int x/((x - 1)^2 (x + 2)) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
If `intsqrt((x - 7)/(x - 9)) dx = Asqrt(x^2 - 16x + 63) + log|x - 8 + sqrt(x^2 - 16x + 63)| + c`, then A = ______
Evaluate the following:
`int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`
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)
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 ______.
Evaluate: `int (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`
Find: `int x^4/((x - 1)(x^2 + 1))dx`.
Find : `int (2x^2 + 3)/(x^2(x^2 + 9))dx; x ≠ 0`.
Evaluate.
`int (5x^2 - 6x + 3) / (2x -3) dx`
Evaluate:
`int (x + 7)/(x^2 + 4x + 7)dx`
