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:
`x/((x^2+1)(x - 1))`
Integrate the rational function:
`(3x -1)/(x + 2)^2`
Integrate the rational function:
`(cos x)/((1-sinx)(2 - sin x))` [Hint: Put sin x = t]
Integrate the rational function:
`(2x)/((x^2 + 1)(x^2 + 3))`
Evaluate : `∫(x+1)/((x+2)(x+3))dx`
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 : `((3sin - 2)*cosx)/(5 - 4sin x - cos^2x)`
Integrate the following w.r.t.x:
`x^2/((x - 1)(3x - 1)(3x - 2)`
Evaluate: `int ("x"^2 + "x" - 1)/("x"^2 + "x" - 6)` dx
Evaluate:
`int x/((x - 1)^2(x + 2)) dx`
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` dx
Evaluate: `int 1/("x"("x"^"n" + 1))` dx
`int x^2sqrt("a"^2 - x^6) "d"x`
`int sqrt(4^x(4^x + 4)) "d"x`
If f'(x) = `x - 3/x^3`, f(1) = `11/2` find f(x)
`int 1/(2 + cosx - sinx) "d"x`
`int sin(logx) "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 (3x + 4)/sqrt(2x^2 + 2x + 1) "d"x`
`int (sin2x)/(3sin^4x - 4sin^2x + 1) "d"x`
`int ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1]) "d"x`
`int (5(x^6 + 1))/(x^2 + 1) "d"x` = x5 – ______ x3 + 5x + c
State whether the following statement is True or False:
For `int (x - 1)/(x + 1)^3 "e"^x"d"x` = ex f(x) + c, f(x) = (x + 1)2
`int x/((x - 1)^2 (x + 2)) "d"x`
Evaluate the following:
`int x^2/(1 - x^4) "d"x` put x2 = t
Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)dx`
Find: `int x^4/((x - 1)(x^2 + 1))dx`.
Evaluate`int(5x^2-6x+3)/(2x-3)dx`
Evaluate:
`int x/((x + 2)(x - 1)^2)dx`
If \[\int\frac{2x+3}{(x-1)(x^{2}+1)}\mathrm{d}x\] = \[=\log_{e}\left\{(x-1)^{\frac{5}{2}}\left(x^{2}+1\right)^{a}\right\}-\frac{1}{2}\tan^{-1}x+\mathrm{A}\] where A is an arbitrary constant, then the value of a is
When is a rational function called improper?
Which partial-fraction decomposition is appropriate for \[\frac{\mathrm{p}x+\mathrm{q}}{(x-\mathrm{a})(x-\mathrm{b})}\]?
What is done after long division, before writing the appropriate partial-fraction decomposition?
Why is \[\frac{x^{2}+1}{x^{2}-5x+6}\] not a proper rational function?
Which decomposition is correct for \[\frac{x^{2}+1}{x^{2}-5x+6}\]?
What must be included for a repeated linear factor?
