Example 2. Find $\int \frac{\operatorname{arctg}^{4} x}{1+x^{2}} d x$.
Solution. Among the tabular integrals, there are no formulas containing arctangent in the integrand. Let's try to make it the variable of integration: $\operatorname{arctg} x=u$. Then $d u=d(\operatorname{arctg} x)=$ $=(\operatorname{arctg} x)^{\prime} d x=\frac{d x}{1+x^{2}}$. Now it is clear that the arctangent should be brought under the differential sign:

$\int \frac{\operatorname{arctg}^{4} x}{1+x^{2}} d x=\int \operatorname{arctg}^{4} x d(\operatorname{arctg} x)=\int u^{4} d u=\frac{u^{5}}{5}+C=\frac{1}{5} \operatorname{arctg}^{5} x+C$.

The transition to a new variable is only useful at the beginning, and later this should be avoided: productivity increases if you save on writing out intermediate calculations.