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Question
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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