Advertisements
Advertisements
प्रश्न
Integrate the rational function:
`x/((x-1)(x- 2)(x - 3))`
Advertisements
उत्तर
Let `x/((x - 1)(x - 2)(x - 3))`
`= A/(x - 1) + B/(x - 2) + C/(x - 3)`
⇒ x = A(x - 2) (x - 3) + B(x - 1) (x - 3) + C(x - 1) (x - 2) …(1)
Putting x = 1 in (i), we get
1 = A(1 - 2) (1 - 3)
⇒ A = `1/2`
Putting x = 2 in (i), we get
2 = B (2 - 1) (2 - 3)
⇒ B = - 2
Putting x = 3 in (i), we get
3 = C(3 - 1) (3 - 2)
⇒ C = `3/2`
`therefore x/((x - 1)(x - 2)(x - 3))`
`= 1/(2(x - 1)) - 2/(x - 2) + 3/(2(x - 3))`
`= int x/((x - 1)(x - 2)(x - 3))` dx
`= 1/2 int 1/(x - 1) dx - 2 int 1/(x - 2) dx + 3/2 int 1/(x - 3) dx`
`= 1/2 log (x - 1) - 2 log (x - 2) + 3/2 log (x - 3) + C`
APPEARS IN
संबंधित प्रश्न
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`x/((x -1)^2 (x+ 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:
`1/(x(x^4 - 1))`
Integrate the rational function:
`1/(e^x -1)`[Hint: Put ex = t]
Evaluate : `∫(x+1)/((x+2)(x+3))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 : `(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:
`(6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)`
Integrate the following w.r.t. x : `(5x^2 + 20x + 6)/(x^3 + 2x ^2 + x)`
Integrate the following w.r.t. x: `(1)/(sinx + sin2x)`
Integrate the following w.r.t. x : `(1)/(sin2x + cosx)`
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 : `sec^2x sqrt(7 + 2 tan x - tan^2 x)`
Evaluate:
`int (2x + 1)/(x(x - 1)(x - 4)) dx`.
Evaluate: `int ("3x" - 1)/("2x"^2 - "x" - 1)` dx
`int (sinx)/(sin3x) "d"x`
`int sec^3x "d"x`
`int "e"^(sin^(-1_x))[(x + sqrt(1 - x^2))/sqrt(1 - x^2)] "d"x`
`int (x + sinx)/(1 - cosx) "d"x`
`int (sin2x)/(3sin^4x - 4sin^2x + 1) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
Evaluate the following:
`int x^2/(1 - x^4) "d"x` put x2 = t
Evaluate the following:
`int_"0"^pi (x"d"x)/(1 + sin x)`
Evaluate the following:
`int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x`
Evaluate the following:
`int "e"^(-3x) cos^3x "d"x`
Evaluate: `int (dx)/(2 + cos x - sin x)`
Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`
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(5x^2-6x+3)/(2x-3)dx`
Which expression defines a rational function?
Which decomposition corresponds to \[\frac{\mathrm{p}x^{2}+\mathrm{q}x+\mathrm{r}}{(x-\mathrm{a})(x^{2}+\mathrm{b}x+\mathrm{c})}\]?
Why is \[\frac{x^{2}+1}{x^{2}-5x+6}\] not a proper rational function?
For \[\frac{5x-5}{(x-2)(x-3)}=\frac{\mathrm{A}}{x-2}+\frac{\mathrm{B}}{x-3}\], which equation results after clearing denominators?
Evaluate \[\int\frac{x^{2}+1}{x^{2}-5x+6}\,dx\].
