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Решите задачу:

$1)\; \; \int \frac{\sqrt{x}-2\sqrt[3]{x^2}+1}{\sqrt[4]{x}} dx=\int (x^{\frac{1}{4}}-2x^{\frac{5}{12}}+x^{-\frac{1}{4}} )dx=\\\\=\frac{4x^{\frac{5}{4}}}{5}-2\cdot \frac{12x^{\frac{17}{12}}}{17}+\frac{4x^{\frac{3}{4}}}{3}+C\\\\2)\; \; \int \frac{\sqrt{x^2-1}-\sqrt{x^2+1}}{\sqrt{x^4-1}} dx=\int \frac{\sqrt{x^2-1}-\sqrt{x^2+1}}{\sqrt{(x^2-1)(x^2+1)}}dx=\int \frac{1}{\sqrt{x^2+1}}dx-\int \frac{1}{\sqrt{x^2-1}}dx\\\\=ln|x+\sqrt{x^2-1}|-ln|x+\sqrt{x^2-1}|+C$

$3)\; \; \int x^2(2-3x^2)dx=\int (2x^2-6x^4)dx=\\\\=2\cdot \frac{x^3}{3}-6\cdot \frac{x^5}{5}+C\\\\4)\; \; \int \frac{(1-x)^3}{x\sqrt[3]{x}}dx=\int \frac{1-3x+3x^2-x^3}{x^{\frac{4}{3}}}dx=\\\\=\int (x^{-\frac{4}{3}}-3x^{-\frac{1}{3}}+3x^{\frac{2}{3}}-x^{\frac{5}{3}})dx=-4x^{-\frac{1}{4}}}-9\frac{x^\frac{2}{3}}{2}+9\frac{x^{\frac{5}{3}}}{5}-\\\\-\frac{3x^{\frac{8}{3}}}{8}+C$

$5)\; \; \int e^{x}(1+\frac{e^{-x}}{cos^2x})dx=\int (e^{x}+\frac{1}{cos^2x})dx=e^{x}+tgx+C$