2.1

145

$\int_{ }^{ }\frac{1}{x^3}dx=\int_{ }^{ }x^{-3}dx=1\cdot\frac{1}{-3+1}x^{-3+1}+C$

$=-\frac{1}{2x^{-2}}+C{,}\ C\in\mathbb{R}$

$\int_{ }^{ }\frac{9}{x^2}dx=\int_{ }^{ }9x^{-2}dx=9\cdot\frac{1}{-2+1}x^{-2+1}+C$

$=-9x^{-1}+C=-\frac{9}{x}+C{,}\ C\in\mathbb{R}$

$\int_{ }^{ }\frac{4}{x}dx=\int_{ }^{ }4\cdot\frac{1}{x}dx=4\cdot\int_{ }^{ }\frac{1}{x}dx=4\ln x+C{,}\ C\in\mathbb{R}$

147

$\int_{ }^{ }\frac{x+2}{x}dx=\int_{ }^{ }\frac{x}{x}+\frac{2}{x}dx=\int_{ }^{ }\frac{2}{x}+1dx=\int_{ }^{ }2\cdot\frac{1}{x}+1dx=2\int_{ }^{ }\frac{1}{x}+1dx=2\ln x+x+C{,}\ C\in\mathbb{R}$

$f\left(x\right)=2\ln x+x-e{,}\ x>0$

149

$\int_{ }^{ }\frac{1}{2x^5}dx=\int_{ }^{ }\frac{1}{2}\cdot\frac{1}{x^5}dx=\frac{1}{2}\int_{ }^{ }x^{-5}dx=\frac{1}{2}\cdot\frac{1}{-5+1}x^{-5+1}+C=\frac{1}{2}\cdot\frac{1}{-4}\cdot\frac{1}{x^4}+C$

$=-\frac{1}{8x^4}+C{,}\ C\in\mathbb{R}$

$\int_{ }^{ }x^2\sqrt[]{x}dx=\int_{ }^{ }x^2x^{\frac{1}{2}}dx=\int_{ }^{ }x^{\frac{5}{2}}dx=\frac{1}{\frac{5}{2}+1}x^{\frac{7}{2}}+C$

$=\frac{2}{7}x^{\frac{7}{2}}+C=\frac{2}{7}x^{\frac{6}{2}}x^{\frac{1}{2}}+C=\frac{2}{7}x^3\sqrt[]{x}+C{,}\ C\in\mathbb{R}$

$\int_{ }^{ }\frac{dx}{2\sqrt[]{x}}=\int_{ }^{ }\frac{1}{2\sqrt[]{x}}dx=\int_{ }^{ }\frac{1}{2}\cdot\frac{1}{\sqrt[]{x}}dx=\frac{1}{2}\int_{ }^{ }\frac{1}{x^{\frac{1}{2}}}dx=\frac{1}{2}\cdot\int_{ }^{ }x^{-\frac{1}{2}}dx$

$=\frac{1}{2}\cdot\frac{1}{-\frac{1}{2}+1}x^{-\frac{1}{2}+1}+C=\frac{1}{2}\cdot2x^{\frac{1}{2}}+C=x^{\frac{1}{2}}+C=\sqrt[]{x}+C{,}\ C\in\mathbb{R}$