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