Advertisements
Advertisements
प्रश्न
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` dx
Advertisements
उत्तर
Let I = `int "3x - 2"/(("x + 1")^2("x + 3"))` dx
Let `"3x - 2"/(("x + 1")^2("x + 3")) = "A"/"x + 1" + "B"/("x + 1")^2 + "C"/("x + 3")`
∴ 3x - 2 = (x + 3) [A(x + 1) + B] + C(x + 1)2 ....(i)
Putting x = - 1 in (i), we get
3(- 1) - 2 = (–1 + 3)[A(0) + B] + C(0)
∴ - 5 = 2B
∴ B = -`5/2`
Putting x = - 3 in (i), we get
3(- 3)-2 = 0[A(–3 + 1) + B] + C(–2)2
∴ - 11 = 4C
∴ C = - `11/4`
Putting x = 0 in (i), we get
3(0)- 2 = 3[A(0 + 1) + B] + C(0 + 1)2
∴ - 2 = 3A + 3B + C
∴ - 2 = 3A + 3`(- 5/2) - 11/4`
∴ 3A = –2 + `15/2 + 11/4 = (- 8 + 30 11)/4 = 33/4`
∴ A = `33/4 xx 1/3 = 11/4`
∴ `"3x - 2"/(("x + 1")^2("x + 3")) = (11/4)/"x + 1" + (- 5/2)/("x + 1")^2 + (- 11/4)/"x + 3"`
∴ I = `int ((11/4)/"x + 1" - (5/2)/("x + 1")^2 - ( 11/4)/"x + 3")` dx
`= 11/4 int "dx"/"x + 1" - 5/2 int ("x + 1")^-2 "dx" - 11/4 int "dx"/"x + 3"`
`= 11/4 log |"x + 1"| - 5/2 (- 1/"x + 1") - 11/4 log |"x + 3"|` + c
`= 11/4 [log |"x" + 1| - log |"x" + 3|] + 5/(2("x" + 1))` + c
∴ I = `11/4 log |("x + 1")/("x + 3")| + 5/(2("x + 1"))` + c
APPEARS IN
संबंधित प्रश्न
Integrate the rational function:
`(3x - 1)/((x - 1)(x - 2)(x - 3))`
Integrate the rational function:
`(5x)/((x + 1)(x^2 - 4))`
Integrate the rational function:
`((x^2 +1)(x^2 + 2))/((x^2 + 3)(x^2+ 4))`
Integrate the rational function:
`1/(x(x^4 - 1))`
Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`
Integrate the following w.r.t. x : `(12x + 3)/(6x^2 + 13x - 63)`
Integrate the following w.r.t. x:
`(6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)`
Choose the correct options from the given alternatives :
If `int tan^3x*sec^3x*dx = (1/m)sec^mx - (1/n)sec^n x + c, "then" (m, n)` =
Integrate the following with respect to the respective variable : `(cos 7x - cos8x)/(1 + 2 cos 5x)`
Integrate the following with respect to the respective variable : `cot^-1 ((1 + sinx)/cosx)`
Integrate the following w.r.t.x : `sec^2x sqrt(7 + 2 tan x - tan^2 x)`
Integrate the following w.r.t.x: `(x + 5)/(x^3 + 3x^2 - x - 3)`
Evaluate:
`int x/((x - 1)^2(x + 2)) dx`
State whether the following statement is True or False.
If `int (("x - 1") "dx")/(("x + 1")("x - 2"))` = A log |x + 1| + B log |x - 2| + c, then A + B = 1.
For `int ("x - 1")/("x + 1")^3 "e"^"x" "dx" = "e"^"x"` f(x) + c, f(x) = (x + 1)2.
`int (2x - 7)/sqrt(4x- 1) dx`
`int x^2sqrt("a"^2 - x^6) "d"x`
`int (sinx)/(sin3x) "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 (6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1) "d"x`
`int x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3)) "d"x`
Evaluate:
`int (5e^x)/((e^x + 1)(e^(2x) + 9)) dx`
If f'(x) = `1/x + x` and f(1) = `5/2`, then f(x) = log x + `x^2/2` + ______ + 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
Evaluate `int x log x "d"x`
`int x/((x - 1)^2 (x + 2)) "d"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 ______.
Evaluate:
`int (x + 7)/(x^2 + 4x + 7)dx`
