Неодређени интеграли – метода смене 4

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Пр.13) $\int {tgxdx } $

Пр.14) $\int {{{\cos }^5}x} {\sin ^3}xdx$

Пр.15)   $\int {\frac{{{{\cos }^5}x}}{{\sqrt {\sin x} }}} dx$

Пр.16)   $\int {{{\sin }^5}\left( {\sin x} \right)} \cos xdx$

Пр.13) $\int {tgxdx = \int {\frac{{\sin x}}{{\cos x}}} } dx = $

Смена: $\cos x = t|'$

$ - \sin x = dt$

$\sin x =  - dt$

$ = \int {\frac{{ - dt}}{t}}  =  - \int {\frac{{dt}}{t}}  =  - \ln \left| t \right| + C =  - \ln \left| {\cos x} \right| + C$

Пр.14) $\int {{{\cos }^5}x} {\sin ^3}xdx = \int {{{\cos }^5}x} {\sin ^2}x\sin xdx = \int {{{\cos }^5}x} \left( {1 - {{\cos }^2}x} \right)\sin xdx = $

Смена: $\cos x = t|'$

$ - \sin x = dt$

$\sin x =  - dt$

$ = \int {{t^5}} \left( {1 - {t^2}} \right)\left( { - dt} \right) =  - \int {\left( {{t^5} - {t^7}} \right)} dt =  - \left( {\int {{t^5}} dt - \int {{t^7}} dt} \right) = $

$ =  - \left( {\frac{{{t^6}}}{6} - \frac{{{t^8}}}{8}} \right) + C = \frac{{{{\cos }^6}x}}{6} - \frac{{{{\cos }^8}x}}{8} + C$

Пр.15)   $\int {\frac{{{{\cos }^5}x}}{{\sqrt {\sin x} }}} dx = \int {\frac{{{{\cos }^4}x\cos x}}{{\sqrt {\sin x} }}} dx = \int {\frac{{{{\left( {{{\cos }^2}x} \right)}^2}\cos x}}{{\sqrt {\sin x} }}} dx = $

$ = \int {\frac{{{{\left( {1 - si{n^2}x} \right)}^2}\cos x}}{{\sqrt {\sin x} }}} dx = $

Смена: $\sin x = t$

$\cos x = dt$

$ = \int {\frac{{{{\left( {1 - {t^2}} \right)}^2}}}{{\sqrt t }}} dt = \int {\frac{{1 - 2{t^2} + {t^4}}}{{\sqrt t }}} dt = \int {\frac{1}{{\sqrt t }}} dt - \int {\frac{{2{t^2}}}{{\sqrt t }}} dt + \int {\frac{{{t^4}}}{{\sqrt t }}} dt = $

$ = \int {{t^{ - \frac{1}{2}}}} dt - 2\int {{t^{\frac{3}{2}}}dt}  + \int {{t^{\frac{7}{2}}}} dt = \frac{{{t^{\frac{1}{2}}}}}{{\frac{1}{2}}} - 2 = 2\sqrt t  - 2\frac{{{t^{\frac{5}{2}}}}}{{\frac{5}{2}}} + \frac{{{t^{\frac{9}{2}}}}}{{\frac{9}{2}}} + C = $

$ = 2\sqrt t  - 2 \cdot \frac{2}{5}{\sqrt t ^5} + \frac{2}{9}{\sqrt t ^9} + C = 2\sqrt {\sin x}  - 2 \cdot \frac{2}{5}{\sqrt {\sin x} ^5} + \frac{2}{9}{\sqrt {\sin x} ^9} + C$

Пр.16)   $\int {{{\sin }^5}\left( {\sin x} \right)} \cos xdx=$

Смена: $\sin x = t$

$\cos x = dt$

$ = \int {{{\sin }^5}t} dt = \int {sint{{\sin }^4}t} dt = \int {sint{{\left( {{{\sin }^2}t} \right)}^2}} dt = \int {sint\left( {1 - {{\cos }^2}t} \right)} dt = $

Смена: $\cos t = s$

$ - \sin tdt = ds$

$\sin tdt =  - ds$

$ = {\int {\left( {1 - {s^2}} \right)} ^2}ds = \int {\left( {1 - 2{s^2} + {s^4}} \right)} ds = \int {ds - 2\int {{s^2}} } ds + \int {{s^4}} ds = $

$ = s - 2\frac{{{s^3}}}{3} + \frac{{{s^5}}}{5} + C = \cos t - 2\frac{{{{\cos }^3}t}}{3} + \frac{{{{\cos }^5}t}}{5} + C = $

$ = \cos \left( {\sin x} \right) - 2\frac{{{{\cos }^3}\left( {\sin x} \right)}}{3} + \frac{{{{\cos }^5}\left( {\sin x} \right)}}{5} + C$