Advertisements
Advertisements
प्रश्न
Evaluate: `int (5"x"^2 + 20"x" + 6)/("x"^3 + 2"x"^2 + "x")` dx
Advertisements
उत्तर
Let I = `int (5"x"^2 + 20"x" + 6)/("x"^3 + 2"x"^2 + "x")` dx
`= int (5"x"^2 + 20"x" + 6)/("x"("x"^2 + 2"x" + 1))` dx
`= int (5"x"^2 + 20"x" + 6)/("x"("x + 1")^2)` dx
Let `(5"x"^2 + 20"x" + 6)/("x"("x + 1")^2) = "A"/"x" + "B"/"x + 1" + "C"/("x + 1")^2`
∴ 5x2 + 20x + 6 = A(x + 1)2 + B(x + 1)x + Cx ...(i)
Putting x = 0 in (i), we get
5(0) + 20(0) + 6 = A(1)2 + B(1)(0) + C(0)
∴ A = 6
Putting x = - 1 in (i), we get
5 (1) + 20(- 1) + 6 = A (0)+ B (0) (- 1) + C (-1)
∴ - 9 = - C
∴ C = 9
Putting x = 1 in (i), we get
5 (1) + 20 (1) + 6 = A (2)2 + B (2) (1) + C (1)
∴ 31 = 4A + 2B + C
∴ 31 = 4(6) + 2B + 9
∴ B = - 1
∴ `(5"x"^2 + 20"x" + 6)/("x"("x + 1")^2) = 6/"x" + (-1)/"x + 1" + 9/("x + 1")^2`
∴ I = `int [6/"x" + (- 1)/"x + 1" + 9/("x + 1")^2]` dx
`= 6 int 1/"x" "dx" - int 1/"x + 1" "dx" + 9 int ("x + 1")^-2` dx
`= 6 log |"x"| - log |"x + 1"| + 9("x + 1")^-1/(-1)` + c
∴ I = `6 log |"x"| - log |"x + 1"| - 9/("x + 1")` + c
APPEARS IN
संबंधित प्रश्न
Evaluate : `int x^2/((x^2+2)(2x^2+1))dx`
Find: `I=intdx/(sinx+sin2x)`
Integrate the rational function:
`x/((x-1)(x- 2)(x - 3))`
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
Integrate the rational function:
`x/((x^2+1)(x - 1))`
`int (xdx)/((x - 1)(x - 2))` equals:
Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`
Find `int (2cos x)/((1-sinx)(1+sin^2 x)) dx`
Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`
Integrate the following w.r.t. x : `(3x - 2)/((x + 1)^2(x + 3)`
Integrate the following w.r.t. x : `(1)/(x^3 - 1)`
Integrate the following with respect to the respective variable : `(6x + 5)^(3/2)`
Integrate the following w.r.t. x: `(x^2 + 3)/((x^2 - 1)(x^2 - 2)`
Integrate the following w.r.t.x : `sec^2x sqrt(7 + 2 tan x - tan^2 x)`
`int sqrt((9 + x)/(9 - x)) "d"x`
`int (sinx)/(sin3x) "d"x`
`int (x^2 + x -1)/(x^2 + x - 6) "d"x`
`int (6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1) "d"x`
`int ("d"x)/(2 + 3tanx)`
`int (x + sinx)/(1 - cosx) "d"x`
Choose the correct alternative:
`int sqrt(1 + x) "d"x` =
If f'(x) = `1/x + x` and f(1) = `5/2`, then f(x) = log x + `x^2/2` + ______ + c
Evaluate `int x log x "d"x`
Evaluate `int x^2"e"^(4x) "d"x`
If `intsqrt((x - 7)/(x - 9)) dx = Asqrt(x^2 - 16x + 63) + log|x - 8 + sqrt(x^2 - 16x + 63)| + c`, then A = ______
Evaluate the following:
`int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`
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 = ______.
Find: `int x^4/((x - 1)(x^2 + 1))dx`.
