Example 8. Find $\int\left(x^{2}+2\right)(\sqrt{x}-3) d x$
Solution. We will expand the brackets and apply the linearity property, i.e., integrate term by term, factoring out numerical coefficients from the integral. In the intermediate integrals, we will use fractional exponents, and express the answer in radicals (roots). We have:

$$
\begin{aligned}
& \int\left(x^{2}+2\right)(\sqrt{x}-3) d x=\int\left(x^{5 / 2}-3 x^{2}+2 x^{\mathrm{I} / 2}-6\right) d x= \\
& \quad=\frac{2}{7} x^{7 / 2}-x^{3}+2 \cdot \frac{2}{3} x^{3 / 2}-6 x+C=\frac{2}{7} \sqrt{x^{7}}-x^{3}+\frac{4}{3} \sqrt{x^{3}}-6 x+C
\end{aligned}
$$

Let's recall that division by a fraction is equivalent to multiplication by its reciprocal.

Further integration of non-tabular integrals is associated with the need to transform the integrands.

$$
\text { Example 9. Find } \int \frac{d x}{\sqrt{5-x}+\sqrt{10-x}} \text {. }
$$

Solution. After multiplying and dividing the fraction by the expression conjugate to the denominator, we obtain tabular integrals

$\int \frac{d x}{\sqrt{5-x}+\sqrt{10-x}}=\int \frac{\sqrt{5-x}-\sqrt{10-x}}{\sqrt{5-x}-\sqrt{10-x}} \cdot \frac{d x}{\sqrt{5-x}+\sqrt{10-x}}=$ $=-\frac{1}{5} \int(\sqrt{5-x}-\sqrt{10-x}) d x=-\frac{2}{15}\left(\sqrt{(5-x)^{3}}-\sqrt{(10-x)^{3}}\right)+C$.