Advertisements
Advertisements
प्रश्न
Evaluate: `int 1/("x"("x"^"n" + 1))` dx
Advertisements
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:
`(3x - 1)/((x - 1)(x - 2)(x - 3))`
Integrate the rational function:
`x/((x -1)^2 (x+ 2))`
Integrate the rational function:
`(3x + 5)/(x^3 - x^2 - x + 1)`
Integrate the rational function:
`(3x -1)/(x + 2)^2`
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:
`(cos x)/((1-sinx)(2 - sin x))` [Hint: Put sin x = t]
Integrate the rational function:
`((x^2 +1)(x^2 + 2))/((x^2 + 3)(x^2+ 4))`
`int (dx)/(x(x^2 + 1))` equals:
Evaluate : `∫(x+1)/((x+2)(x+3))dx`
Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`
Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`
Integrate the following w.r.t. x:
`(6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)`
Integrate the following w.r.t. x : `(3x - 2)/((x + 1)^2(x + 3)`
Integrate the following w.r.t. x : `(5x^2 + 20x + 6)/(x^3 + 2x ^2 + x)`
Integrate the following w.r.t. x : `(1)/(sinx*(3 + 2cosx)`
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 : `(cos 7x - cos8x)/(1 + 2 cos 5x)`
Integrate the following w.r.t.x : `(1)/(sinx + sin2x)`
Integrate the following w.r.t.x: `(x + 5)/(x^3 + 3x^2 - x - 3)`
Evaluate: `int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10)` dx
Evaluate: `int (1 + log "x")/("x"(3 + log "x")(2 + 3 log "x"))` dx
`int sin(logx) "d"x`
`int "e"^(sin^(-1_x))[(x + sqrt(1 - x^2))/sqrt(1 - x^2)] "d"x`
`int ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1]) "d"x`
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 ______.
Evaluate:
`int 2/((1 - x)(1 + x^2))dx`
Evaluate.
`int (5x^2 - 6x + 3) / (2x -3) dx`
Value of ∫ `(x^2 + 1)/((x − 1)(x − 2))`dx is ______.
