Question

Evaluate: `int 1/("x"("x"^"n" + 1))` dx

Answer 1

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