Evaluate the following. ∫x5x2+1dx - Mathematics and Statistics

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Sum

Evaluate the following.

`int "x"^5/("x"^2 + 1)`dx

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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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Chapter 5: Integration - EXERCISE 5.2 [Page 123]

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