Advertisements
Advertisements
प्रश्न
Evaluate: `int 1/("x"("x"^"n" + 1))` dx
उत्तर
Let I = `int 1/("x"("x"^"n" + 1))` dx
∴ I = `int "x"^"n - 1"/("x"^"n - 1" xx "x"("x"^"n" + 1))` dx
∴ I = `int "x"^"n - 1"/("x"^"n" ("x"^"n" + 1))` dx
Put xn = t
∴ `"n""x"^"n - 1" "dx" = "dt"`
∴ `"x"^"n - 1" "dx" = "dt"/"n"`
∴ I = `int 1/("t"("t + 1")) * "dt"/"n"`
Let `1/("t"("t + 1")) = "A"/"t" + "B"/"t + 1"`
∴ 1 = A(t + 1) + Bt ....(i)
Putting t = –1 in (i), we get
1 = A(0) + B(- 1)
∴ 1 = - B
∴ B = - 1
Putting t = 0 in (i), we get
1 = A(1) + B(0)
∴ A = 1
∴ `1/("t"("t + 1")) = 1/"t" + (- 1)/"t + 1"`
∴ I = `1/"n" int (1/"t" + (-1)/"t + 1")` dt
`= 1/"n" [int 1/"t" "dt" - int 1/("t + 1") "dt"]`
`= 1/"n" [log |"t"| - log |"t" + 1|]` + c
`= 1/"n" log |"t"/"t + 1"|` + c
∴ I = `1/"n" log |"x"^"n"/("x"^"n" + 1)|` + c
APPEARS IN
संबंधित प्रश्न
Integrate the rational function:
`1/(x^2 - 9)`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`(3x + 5)/(x^3 - x^2 - x + 1)`
Integrate the rational function:
`(cos x)/((1-sinx)(2 - sin x))` [Hint: Put sin x = t]
Integrate the rational function:
`1/(e^x -1)`[Hint: Put ex = t]
Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`
Integrate the following w.r.t. x : `(12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)`
Integrate the following w.r.t. x : `(1)/(x(x^5 + 1)`
Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
Integrate the following w.r.t.x : `sqrt(tanx)/(sinx*cosx)`
Evaluate: `int (5"x"^2 + 20"x" + 6)/("x"^3 + 2"x"^2 + "x")` dx
Evaluate: `int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10)` dx
`int (6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1) "d"x`
`int x^3tan^(-1)x "d"x`
Evaluate the following:
`int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`
If `int "dx"/((x + 2)(x^2 + 1)) = "a"log|1 + x^2| + "b" tan^-1x + 1/5 log|x + 2| + "C"`, then ______.
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 dx/sqrt(16 - 9x^2)` = A sin–1 (Bx) + C then A + B = ______.
Find: `int x^4/((x - 1)(x^2 + 1))dx`.
Evaluate:
`int 2/((1 - x)(1 + x^2))dx`
