Неодређени интеграли 2

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Пр.9)      $\int {\frac{{\sin \frac{\pi }{6}}}{{{{\sin }^2}x}}} dx$

Пр.10)    $\int {\left( {8{x^3} - 3{x^2}} \right)} dx$

Пр.11)    $\int {\frac{{{{\left( {1 - {x^2}} \right)}^2}}}{{{x^3}}}} dx$

Пр.12)    $\int {\frac{{\left( {x - \sqrt x } \right)\left( {1 - \sqrt x } \right)}}{{\sqrt[3]{x}}}} dx$

Пр.9)      $\int {\frac{{\sin \frac{\pi }{6}}}{{{{\sin }^2}x}}} dx = \int {\frac{{\frac{1}{2}}}{{{{\sin }^2}x}}} dx = \frac{1}{2}\int {\frac{{dx}}{{{{\sin }^2}x}}}  =  - \frac{1}{2}ctgx + C$

Пр.10)    $\int {\left( {8{x^3} - 3{x^2}} \right)} dx = \int {8{x^3}} dx - \int {3{x^2}} dx = 8\int {{x^3}} dx - 3\int {{x^2}} dx = 8 \cdot \frac{{{x^4}}}{4} - 3 \cdot \frac{{{x^3}}}{3} + C =$

$=8 \cdot \frac{{{x^4}}}{4} - 3 \cdot \frac{{{x^3}}}{3} + C =2{x^4} - {x^3} + C$

Пр.11)    $\int {\frac{{{{\left( {1 - {x^2}} \right)}^2}}}{{{x^3}}}} dx = \int {\frac{{1 - 2{x^2} + {x^4}}}{{{x^3}}}} dx = \int {(\frac{1}{{{x^3}}}}  - \frac{{2{x^2}}}{{{x^3}}} + \frac{{{x^4}}}{{{x^3}}})dx = $

$ = \int {{x^{ - 3}}} dx - \int {\frac{2}{x}} dx + \int {xdx}  = \int {{x^{ - 3}}} dx - 2\int {\frac{1}{x}} dx + \int {xdx}  = $

$ = \frac{{{x^{ - 2}}}}{{ - 2}} - 2\ln x + \frac{{{x^2}}}{2}\int {{x^{ - 3}}}  + c$

Пр.12)    $\int {\frac{{\left( {x - \sqrt x } \right)\left( {1 - \sqrt x } \right)}}{{\sqrt[3]{x}}}} dx = \int {\frac{{x - x\sqrt x  - \sqrt x  + x}}{{\sqrt[3]{x}}}} dx = \int {\frac{{2x - x \cdot {x^{\frac{1}{2}}} - {x^{\frac{1}{2}}}}}{{{x^{\frac{1}{3}}}}}} dx = $

$ = \int {\left( {\frac{{2x}}{{{x^{\frac{1}{3}}}}} - \frac{{{x^{\frac{3}{2}}}}}{{{x^{\frac{1}{3}}}}} - \frac{{{x^{\frac{1}{2}}}}}{{{x^{\frac{1}{3}}}}}} \right)} dx = \int {\frac{{2x}}{{{x^{\frac{1}{3}}}}}dx - } \int {\frac{{{x^{\frac{3}{2}}}}}{{{x^{\frac{1}{3}}}}}dx - } \int {\frac{{{x^{\frac{1}{2}}}}}{{{x^{\frac{1}{3}}}}}} dx = \int {2{x^{\frac{2}{3}}}}  - {x^{\frac{7}{6}}} - {x^{\frac{1}{6}}}dx = $

$ = \int {2{x^{\frac{2}{3}}}} dx - \int {{x^{\frac{7}{6}}}} dx - \int {{x^{\frac{1}{6}}}dx}  = 2\frac{{{x^{\frac{2}{3} + 1}}}}{{^{\frac{2}{3} + 1}}} - \frac{{{x^{\frac{7}{6} + 1}}}}{{\frac{7}{6} + 1}} - \frac{{{x^{\frac{1}{6} + 1}}}}{{\frac{1}{6} + 1}} + C = $

$ = 2\frac{{{x^{\frac{5}{3}}}}}{{\frac{5}{3}}} - \frac{{{x^{\frac{{13}}{6}}}}}{{\frac{{13}}{6}}} - \frac{{{x^{\frac{7}{6}}}}}{{\frac{7}{6}}} + C = \frac{6}{5}{x^{\frac{5}{3}}} - \frac{6}{{13}}{x^{\frac{{13}}{6}}} - \frac{6}{7}{x^{\frac{7}{6}}} + C$