51. $\int \frac{x^{3}+1}{x} d x$.

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51. $\int \frac{x^{3}+1}{x} d x$.
Solution. Dividing the numerator term by term by $x$, we represent the integrand as the sum of two fractions:

$$
\int \frac{x^{3}+1}{x} d x=\int\left(x^{2}+\frac{1}{x}\right) d x
$$

We break the last integral into the sum of two integrals and apply formulas I and II. Then we get

$$
\int\left(x^{2}+\frac{1}{x}\right) d x=\int x^{2} d x+\int \frac{1}{x} d x=\frac{x^{3}}{3}+\ln |x|+C
$$

By differentiating the result, it is easy to verify that the solution is correct.