Advertisements
Advertisements
प्रश्न
Integrate the rational function:
`(3x - 1)/((x - 1)(x - 2)(x - 3))`
Advertisements
उत्तर
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
संबंधित प्रश्न
Integrate the rational function:
`2/((1-x)(1+x^2))`
Integrate the rational function:
`1/(x^4 - 1)`
Integrate the rational function:
`1/(x(x^n + 1))` [Hint: multiply numerator and denominator by xn − 1 and put xn = t]
Integrate the rational function:
`1/(x(x^4 - 1))`
Integrate the following w.r.t. x : `(2x)/((2 + x^2)(3 + x^2)`
Integrate the following w.r.t. x : `(5x^2 + 20x + 6)/(x^3 + 2x ^2 + x)`
Integrate the following w.r.t. x : `(5*e^x)/((e^x + 1)(e^(2x) + 9)`
Integrate the following with respect to the respective variable : `(6x + 5)^(3/2)`
Integrate the following w.r.t.x : `sec^2x sqrt(7 + 2 tan x - tan^2 x)`
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` dx
`int "dx"/(("x" - 8)("x" + 7))`=
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 sqrt((9 + x)/(9 - x)) "d"x`
`int sec^2x sqrt(tan^2x + tanx - 7) "d"x`
`int "e"^(sin^(-1_x))[(x + sqrt(1 - x^2))/sqrt(1 - x^2)] "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 (x + sinx)/(1 - cosx) "d"x`
Evaluate:
`int (5e^x)/((e^x + 1)(e^(2x) + 9)) dx`
`int 1/(sinx(3 + 2cosx)) "d"x`
`int (sin2x)/(3sin^4x - 4sin^2x + 1) "d"x`
`int ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1]) "d"x`
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
Evaluate `int x^2"e"^(4x) "d"x`
`int x/((x - 1)^2 (x + 2)) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
Evaluate the following:
`int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`
Evaluate the following:
`int_"0"^pi (x"d"x)/(1 + sin x)`
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 ______.
If `int 1/((x^2 + 4)(x^2 + 9))dx = A tan^-1 x/2 + B tan^-1(x/3) + C`, then A – B = ______.
Evaluate: `int (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`
Find: `int x^4/((x - 1)(x^2 + 1))dx`.
Find : `int (2x^2 + 3)/(x^2(x^2 + 9))dx; x ≠ 0`.
Evaluate:
`int x/((x + 2)(x - 1)^2)dx`
When is a rational function \[\frac{P(x)}{Q(x)}\] called proper?
Which partial-fraction decomposition is appropriate for \[\frac{\mathrm{p}x+\mathrm{q}}{(x-\mathrm{a})(x-\mathrm{b})}\]?
