Evaluate the integral of 1 / (x * ($x^4 + 1$)) dx.

Step 1: Rewrite the integrand

To simplify the integral, we multiply the numerator and denominator by $x^3$. This allows us to make a substitution involving $x^4$ later.

Step 2: Substitute $u = x^4$

We introduce a substitution to simplify the integral further. Let $u = x^4$. Then, we find the differential du by differentiating u with respect to x. This allows us to replace $x^3 dx$ with $\frac{1}{4} du$.

Step 3: Substitute into the integral

Now we substitute u and du into the integral. The integral becomes $\frac{1}{4} \int \frac{1}{u(u + 1)} du$.

Step 4: Use partial fraction decomposition

We use partial fraction decomposition to break down the integrand $\frac{1}{u(u + 1)}$ $. We set \frac{1}{u(u + 1)} = \frac{A}{u} + \frac{B}{u + 1}$. By solving for A and B, we find $A = 1$ and $B = -1$.

Step 5: Integrate with respect to $u$

Now we integrate the decomposed fractions. The integral of $\frac{1}{u}$ is $\ln|u|$, and the integral of $\frac{1}{u + 1}$ is $\ln|u + 1|$. We combine the logarithms using the property $\ln a - \ln b = \ln\left(\frac{a}{b}\right)$.

Step 6: Substitute back $x^4$ for $u$

Finally, we substitute $x^4$ back in for u to express the result in terms of x.

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