Question

Найти интеграл на фото​

Answer 1

$\int \frac{\sqrt{x}}{1+\sqrt[4]{x}}\, dx=\Big [\; NOK(2,4)=4,\; x=t^4,\; dx=4x^3\, dx,\; \sqrt{x}=\sqrt{t^4}=t^2,\\\\\sqrt[4]{x}=\sqrt[4]{t^4}=t\; ,\; t=\sqrt[4]{x}\; \Big ]=\int \frac{t^2\cdot 4t^3\, dt}{1+t}=4\int \frac{t^5\, dt}{t+1}=\\\\=\int (t^4-t^3+t^2-t+1-\frac{1}{t+1})\, dt=\\\\=\frac{t^5}{5}-\frac{t^4}{4}+\frac{t^3}{3}-\frac{t^2}{2}+t-ln|t+1|+C=\\\\=\frac{\sqrt[4]{x^5}}{5}-\frac{x}{4}+\frac{\sqrt[4]{x}^3}{3}-\frac{\sqrt{x}}{2}+\sqrt[4]{x}-ln|\sqrt[4]{x}+1|+C$