Задача №1527

\[ \int\frac{3+x^3}{\sqrt{2+2x^2}}dx =\frac{3}{\sqrt{2}}\int\frac{dx}{\sqrt{x^2+1}}+\frac{1}{\sqrt{2}}\int\frac{x^3dx}{\sqrt{x^2+1}} =\frac{3}{\sqrt{2}}\ln\left(x+\sqrt{1+x^2}\right)+\frac{1}{2\sqrt{2}}\int\frac{x^3\cdot{2xdx}}{\sqrt{x^2+1}}=\\ =\left[t=x^2+1;\;dt=2xdx.\right] =\frac{3}{\sqrt{2}}\ln\left(x+\sqrt{1+x^2}\right)+\frac{1}{2\sqrt{2}}\int\left(t^{\frac{1}{2}}-t^{-\frac{1}{2}}\right)dt=\\ =\frac{3}{\sqrt{2}}\ln\left(x+\sqrt{1+x^2}\right)+\frac{1}{2\sqrt{2}}\left(\frac{2t\sqrt{t}}{3}-2\sqrt{t}\right)+C =\frac{3}{\sqrt{2}}\ln\left(x+\sqrt{1+x^2}\right)+\frac{1}{3\sqrt{2}}\left(x^2-2\right)\sqrt{x^2+1}+C \]