Question

Integrate the following w.r.t. x : `(1)/(x(x^5 + 1)`

Answer 1

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`.