Question

Evaluate: `int 1/("x"("x"^5 + 1))` dx

Answer 1

Let I = `int 1/("x"("x"^5 + 1))` dx

∴ I = `int "x"^4/("x"^5("x"^5 + 1))` dx

Put x⁵ = t

∴ `5"x"^4  "dx" = "dt"`

∴ `"x"^4  "dx" = "dt"/5`

∴ I = `int 1/("t"("t + 1")) * "dt"/5`

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 = `1/5 int (1/"t" + (-1)/"t + 1")` dt

`= 1/5 [int 1/"t" "dt" - int 1/("t + 1") "dt"]`

`= 1/5 [log |"t"| - log |"t" + 1|]` + c

`= 1/5 log |"t"/"t + 1"|` + c

∴ I = `1/5 log |"x"^5/("x"^5 + 1)|` + c