Advertisements
Advertisements
Question
Integrate the rational function:
`(3x - 1)/((x - 1)(x - 2)(x - 3))`
Advertisements
Solution
Let `(3x - 1)/((x - 1)(x - 2)(x - 3))`
`= A/(x - 1) + B/(x - 2) + C/(x - 3)`
⇒ 3x - 1 = A(x - 2) (x - 3) + B(x - 1) (x - 3) + C(x - 1) (x - 2) …(1)
Putting x = 1 in (i), we get
3 - 1 = A(1 - 2) (1 - 3)
⇒ 2 = A(-1) (-2)
⇒ A = 1
Putting x = 2 in (i), we get
6 - 1 = B (2 - 1) (2 - 3)
⇒ 5 = B(1) (-1)
⇒ B = -5
Putting x = 3 in (i), we get
9 - 1 = C (3 - 1) (3 - 2)
⇒ 8 = C (2) (1)
⇒ C = 4
`therefore (3x - 1)/((x - 1)(x - 2)(x - 3))`
`= 1/(x - 1) - 5/(x - 2) + 4/(x - 3)`
`= int (3x - 1)/((x - 1)(x - 2)(x - 3))` dx
`= int1/(x - 1) dx - 5 int 1/(x - 2) dx + 4 int 1/(x - 3) dx`
= log (x - 1) - 5 log (x - 2) + 4 log (x - 3) + C
APPEARS IN
RELATED QUESTIONS
Evaluate: `∫8/((x+2)(x^2+4))dx`
Integrate the rational function:
`x/((x-1)(x- 2)(x - 3))`
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
Integrate the rational function:
`(2x)/((x^2 + 1)(x^2 + 3))`
`int (xdx)/((x - 1)(x - 2))` equals:
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 : `(2log x + 3)/(x(3 log x + 2)[(logx)^2 + 1]`
Integrate the following with respect to the respective variable : `(6x + 5)^(3/2)`
Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
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 : `cot^-1 ((1 + sinx)/cosx)`
Integrate the following w.r.t.x : `x^2/sqrt(1 - x^6)`
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` dx
`int (2x - 7)/sqrt(4x- 1) dx`
If f'(x) = `x - 3/x^3`, f(1) = `11/2` find f(x)
`int (7 + 4x + 5x^2)/(2x + 3)^(3/2) dx`
`int 1/(2 + cosx - sinx) "d"x`
`int "e"^(sin^(-1_x))[(x + sqrt(1 - x^2))/sqrt(1 - x^2)] "d"x`
`int x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3)) "d"x`
`int 1/(sinx(3 + 2cosx)) "d"x`
`int xcos^3x "d"x`
`int (3"e"^(2x) + 5)/(4"e"^(2x) - 5) "d"x`
Choose the correct alternative:
`int sqrt(1 + x) "d"x` =
`int (5(x^6 + 1))/(x^2 + 1) "d"x` = x5 – ______ x3 + 5x + c
`int (3"e"^(2"t") + 5)/(4"e"^(2"t") - 5) "dt"`
Evaluate the following:
`int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`
Evaluate the following:
`int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x`
The numerator of a fraction is 4 less than its denominator. If the numerator is decreased by 2 and the denominator is increased by 1, the denominator becomes eight times the numerator. Find the fraction.
Evaluate: `int (dx)/(2 + cos x - sin x)`
Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)dx`
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 ______.
When is a rational function \[\frac{P(x)}{Q(x)}\] called proper?
A proper rational function can be expressed as a sum of simpler rational functions called what?
Which partial form corresponds to \[\frac{\mathrm{p}x^{2}+\mathrm{q}x+\mathrm{r}}{(x-\mathrm{a})(x-\mathrm{b})(x-\mathrm{c})}\]?
