Advertisements
Advertisements
Question
Integrate the rational function:
`(5x)/((x + 1)(x^2 - 4))`
Solution
`(5x)/((x + 1)(x^2 - 4))`
`= 5/((x + 1)(x + 2)(x - 2))`
`(5x)/((x + 1)(x^2 - 4)) => A/(x + 1) + B/(x + 2) + C/(x - 2)`
⇒ 5x = A(x2 - 4) + B (x + 1)(x - 2) + C(x + 1)(x + 2)
Put x = -1
-5 = -3A + 0 = 0
⇒ A `= 5/3`
Put x = -2
-10 = 0 + B(-1)(-4) + 0
⇒ B `= (-5)/2`
Put x = 2
10 = 0 + 0 + 12C
⇒ C `= 5/6`
`therefore int (5x)/((x + 1)(x^2 - 4)`
`= 5/3 int 1/(x + 1) dx - 5/2 int 1/(x + 1) dx + 5/6 int 1/(x - 2) dx`
`= 5/3 log abs (x + 1) - 5/2 log abs (x + 1) + 5/6 log abs (x - 2) + C`
APPEARS IN
RELATED QUESTIONS
Find : `int x^2/(x^4+x^2-2) dx`
Find: `I=intdx/(sinx+sin2x)`
Integrate the rational function:
`(3x - 1)/((x - 1)(x - 2)(x - 3))`
Integrate the rational function:
`(x^3 + x + 1)/(x^2 -1)`
Integrate the rational function:
`(3x -1)/(x + 2)^2`
Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`
Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`
Integrate the following w.r.t. x : `(1)/(sinx*(3 + 2cosx)`
Integrate the following w.r.t. x: `(x^2 + 3)/((x^2 - 1)(x^2 - 2)`
Integrate the following w.r.t.x : `(1)/(2cosx + 3sinx)`
Evaluate: `int "x"/(("x - 1")^2("x + 2"))` dx
Evaluate: `int (5"x"^2 + 20"x" + 6)/("x"^3 + 2"x"^2 + "x")` dx
`int "dx"/(("x" - 8)("x" + 7))`=
Evaluate: `int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10)` dx
`int x^7/(1 + x^4)^2 "d"x`
`int (7 + 4x + 5x^2)/(2x + 3)^(3/2) dx`
`int ("d"x)/(x^3 - 1)`
`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` =
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
Evaluate `int (2"e"^x + 5)/(2"e"^x + 1) "d"x`
`int (3"e"^(2"t") + 5)/(4"e"^(2"t") - 5) "dt"`
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 sqrt(tanx) "d"x` (Hint: Put tanx = t2)
Evaluate: `int (dx)/(2 + cos x - sin x)`
Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`
Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)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 ______.
Find: `int x^4/((x - 1)(x^2 + 1))dx`.
Evaluate:
`int x/((x + 2)(x - 1)^2)dx`
