$$ 31.\,\int\!\! \frac{dx}{\sqrt{x^2+a^2}} = arsh \frac{x}{a}+C1= \ln (x+\sqrt{x^2+a^2})+C $$

$$ 32.\,\int\!\! \frac{dx}{\sqrt{(x^2+a^2)^3}}= \frac{x}{a^2\sqrt{x^2+a^2}}+C $$

$$ 33.\,\int\!\! \frac{x}{\sqrt{x^2+a^2}}dx=\sqrt{x^2+a^2}+C $$

$$ 34.\,\int\!\! \frac{x}{\sqrt{(x^2+a^2)^3}}dx= - \frac{1}{\sqrt{x^2+a^2}}+C $$

$$ 35.\,\int\!\! \frac{x^2}{\sqrt{x^2+a^2}}dx = \frac{x}{2}\sqrt{x^2+a^2}-\frac{a^2}{2}\ln(x+\sqrt{x^2+a^2})+C $$

$$ 36.\,\int\!\! \frac{x^2}{\sqrt{(x^2+a^2)^3}}dx= -\frac{x}{\sqrt{x^2+a^2}}+\ln (x+\sqrt{x^2+a^2})+C $$

$$ 37.\,\int\!\! \frac{dx}{x\sqrt{x^2+a^2}}= \frac{1}{a}\ln \frac{\sqrt{x^2+a^2}-a}{\vert x \vert} +C $$

$$ 38.\,\int\!\! \frac{dx}{x^2\sqrt{x^2+a^2}}= -\frac{\sqrt{x^2+a^2}}{a^2x}+C $$

$$ 39.\,\int\!\! \sqrt{x^2+a^2}dx=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\ln(x+\sqrt{x^2+a^2})+C $$

$$ 40.\,\int\!\! \sqrt{(x^2+a^2)^3}dx = \frac{x}{8}(2x^2+5a^2)\sqrt{x^2+a^2}+ \frac{3}{8}a^4\ln(x+\sqrt{x^2+a^2})+C $$

$$ 41.\,\int\!\! x\sqrt{x^2+a^2}dx= \frac{1}{3}\sqrt{(x^2+a^2)^3}+C $$

$$ 42.\,\int\!\! x^2 \sqrt{x^2+a^2}dx=\frac{x}{8}(2x^2+a^2)\sqrt{x^2+a^2}-\frac{a^4}{8}\ln(x+\sqrt{x^2+a^2})+C $$

$$ 43.\,\int\!\! \frac{\sqrt{x^2+a^2}}{x}dx= \sqrt{x^2+a^2}+a\ln\frac{\sqrt{x^2+a^2}-a}{\vert x \vert} +C $$

$$ 44.\,\int\!\! \frac{\sqrt{x^2+a^2}}{x^2}dx= -\frac{\sqrt{x^2+a^2}}{x} + \ln(x+\sqrt{x^2+a^2}) +C $$