Advertisements
Advertisements
प्रश्न
Integrate the rational function:
`x/((x^2+1)(x - 1))`
Advertisements
उत्तर
Let `x/((x^2 + 1)(x - 1)) = (Ax + B)/(x^2 + 1) + C/((x - 1))`
⇒ x = (A) (+ B)(x - 1) + C = `1/2`
Put x = 1
1 = 0 + 2C
⇒ C `= 1/2`
On comparing the coefficients of x2 or x
0 = A + C
⇒ A = `- 1/2`
and 1 = - A + B
⇒ B `= 1/2`
Hence, `int x/((x^2 + 1)(x - 1)) dx`
`= int (- 1/2 x + 1/2)/(x^2 + 1) dx + 1/2 int 1/(x - 1) dx`
`= -1/2 int (x - 1)/(x^2 + 1) dx + 1/2 log abs (x - 1) + C`
`= 1/4 int (2x)/(x^2 + 1) + 1/2 int 1/(x^2 + 1) dx + 1/2 log abs (x - 1) + C`
`= - 1/4 log abs (x^2 + 1) + 1/2 tan^-1 x + 1/2 log abs (x - 1) + C`
APPEARS IN
संबंधित प्रश्न
Evaluate:
`int x^2/(x^4+x^2-2)dx`
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
Integrate the rational function:
`(3x + 5)/(x^3 - x^2 - x + 1)`
Integrate the rational function:
`(2x - 3)/((x^2 -1)(2x + 3))`
Integrate the rational function:
`(x^3 + x + 1)/(x^2 -1)`
Integrate the rational function:
`1/(x(x^4 - 1))`
`int (dx)/(x(x^2 + 1))` equals:
Find `int (2cos x)/((1-sinx)(1+sin^2 x)) dx`
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 : `((3sin - 2)*cosx)/(5 - 4sin x - cos^2x)`
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 : `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 : `(1)/(2cosx + 3sinx)`
Integrate the following w.r.t.x:
`x^2/((x - 1)(3x - 1)(3x - 2)`
Evaluate:
`int (2x + 1)/(x(x - 1)(x - 4)) dx`.
Evaluate: `int ("x"^2 + "x" - 1)/("x"^2 + "x" - 6)` dx
Evaluate: `int 1/("x"("x"^5 + 1))` dx
`int x^2sqrt("a"^2 - x^6) "d"x`
`int (sinx)/(sin3x) "d"x`
Evaluate:
`int (5e^x)/((e^x + 1)(e^(2x) + 9)) dx`
Choose the correct alternative:
`int ((x^3 + 3x^2 + 3x + 1))/(x + 1)^5 "d"x` =
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 `int x^2"e"^(4x) "d"x`
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 (x^2"d"x)/(x^4 - x^2 - 12)`
Evaluate the following:
`int "e"^(-3x) cos^3x "d"x`
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)
If `intsqrt((x - 5)/(x - 7))dx = Asqrt(x^2 - 12x + 35) + log|x| - 6 + sqrt(x^2 - 12x + 35) + C|`, then A = ______.
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 (5x^2 - 6x + 3)/(2x - 3)dx`
