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Question
Integrate the following w.r.t. x : `(1)/(x(x^5 + 1)`
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Solution
Let I = `int (1)/(x(x^5 + 1)).dx`
= `int x^4/(x^5(x^5 + 1)).dx`
Put x5 = t.
Then 5x4 dx = dt
∴ x4 dx= `dt/(5)`
∴ I = `int (1)/(t(t + 1)).dt/(5)`
= `(1)/(5) int ((t + 1) - t)/(t(t + 1)).dt`
= `(1)/(5) int (1/t - 1/(t + 1)).dt`
= `(1)/(5)[ int 1/t dt - int (1)/(t + 1)dt]`
= `(1)/(5)[log|t| - log|t + 1|] + c`
= `(1)/(5)log|t/(t + 1)| + c`
= `(1)/(5)log|x^5/(x^5 + 1)| + c`.
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