Evaluate the following. ∫1x(x6+1) dx - Mathematics and Statistics

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Sum

Evaluate the following.

`int 1/("x"("x"^6 + 1))` dx

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Solution

Let I = `int 1/("x"("x"^6 + 1))` dx

`= int "x"^5/("x"^6("x"^6 + 1))`dx

Put x6 = t

∴ 6x5 dx = dt

∴ `"x"^5 * "dx" = 1/6 * "dt"`

∴ I = `1/6 int "dt"/("t"("t + 1"))`

`= 1/6 int (("t + 1") - "t")/("t"("t" + 1))` dt

`= 1/6 int (1/"t" - 1/("t + 1"))` dt

= `1/6` [log | t | - log |t + 1|] + c

`= 1/6 log |"t"/("t + 1")|` + c

∴ I = `1/6 log |"x"^6/("x"^6 + 1)|` + c

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Chapter 5: Integration - EXERCISE 5.2 [Page 123]

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