Best solutions to - "Evaluate the integral. ∫dxx4−x8"

Barbara Schroder

Answered question

2021-11-11

Evaluate the integral.

\int \frac{d x}{x \sqrt{4 - x^{8}}}

Feas1981

Beginner2021-11-12Added 16 answers

Step 1
Given integral is:

I = \int \frac{d x}{x \sqrt{4 - x^{8}}}

Let us substitute:

\sqrt{4 - x^{8}} = u

Then we get,

I = - \frac{1}{4} \int \frac{d u}{4 - u^{2}}

Step 2
Let u=2v
Then we get,

I = (- \frac{1}{4}) (\frac{1}{2}) \int \frac{d v}{1 - v^{2}}

I = - \frac{1}{8} (\frac{\ln | v + 1 |}{2} - \frac{\ln | v - 1 |}{2}) + c

Using back substitution we get,

I = - \frac{1}{16} (\ln | \frac{\sqrt{4 - x^{8}}}{2} + 1 | - \ln | \frac{\sqrt{4 - x^{8}}}{2} - 1 |) + c

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