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Question
Evaluate the following.
`int "x"^5/("x"^2 + 1)`dx
Solution
Let I = `int "x"^5/("x"^2 + 1)`dx
`int (("x"^2)^2 * "x")/("x"^2 + 1)`dx
Put x2 + 1 = t
∴ 2x . dx = dt
∴ x . dx = `1/2 * "dt"`
Also, x2 = t - 1
∴ I = `int ("t" - 1)^2/"t" * 1/2`dt
`= 1/2 int ("t"^2 - 2"t" + 1)/"t"`dt
`= 1/2 int ("t" - 2 + 1/"t")`dt
`= 1/2 ["t"^2/2 - 2"t" + log |"t"|]` + c
`= 1/4 "t"^2 - "t" + 1/2 log |"t"| + "c"`
∴ I = `1/4 ("x"^2 + 1)^2 - ("x"^2 + 1) + 1/2 log |"x"^2 + 1|` + c
Notes
The answer in the textbook is incorrect.
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