Неодређени интеграли - парцијална интеграција 3
Интеграли. Особине интеграла. Парцијална интеграција. Дефиниција. Сложенији примери.
Задаци
Текст задатака објашњених у видео лекцији:
Пр.6) Решити $\int {{x^3}\ln xdx} $
Пр.7) $\int {xarctgdx} $
Пр.8) $\int {\frac{{\operatorname{arcsinx} dx}}{{{{\sqrt {1 - {x^2}} }^3}}}} $
Пр.6) $\int {{x^3}\ln xdx} $
\[\begin{array}{*{20}{c}}
\begin{gathered}
u = \ln x \hfill \\
du = \frac{1}{x}dx \hfill \\
\end{gathered} &{}&\begin{gathered}
dv = {x^3}dx \hfill \\
v = \int {{x^3}dx = \frac{{{x^4}}}{4} + C} \hfill \\
\end{gathered}
\end{array}\]
$ = \frac{{{x^4}}}{4}\ln x - \int {\frac{{{x^4}}}{4}} \cdot \frac{1}{x}dx = \frac{{{x^4}}}{4}\ln x - \frac{1}{4}\int {{x^3}} dx = \frac{{{x^4}}}{4}\ln x - \frac{1}{4} \cdot \frac{{{x^4}}}{4} + C = $
$ = \frac{{{x^4}}}{4}\left( {\ln x - \frac{1}{4}} \right) + C$
Пр.7) $\int {xarctgxdx} = $
\[\begin{array}{*{20}{c}}
\begin{gathered}
u = arctgx \hfill \\
du = \frac{1}{{1 + {x^2}}}dx \hfill \\
\end{gathered} &{}&\begin{gathered}
dv = xdx \hfill \\
v = \int {xdx = \frac{{{x^2}}}{2} + C} \hfill \\
\end{gathered}
\end{array}\]
$ = \frac{{{x^2}}}{2}arctgx - \int {\frac{{{x^2}}}{2} \cdot \frac{1}{{1 + {x^2}}}dx} = \frac{{{x^2}}}{2}arctgx - \frac{1}{2}\int {\frac{{1 + {x^2} - 1}}{{1 + {x^2}}}dx = } $
$ = \frac{{{x^2}}}{2}arctgx - \frac{1}{2}\left( {\int {\frac{{1 + {x^2}}}{{1 + {x^2}}}dx - \int {\frac{1}{{1 + {x^2}}}dx} } } \right) = $
$ = \frac{{{x^2}}}{2}arctgx - \frac{1}{2}\left( {x - arctgx} \right) + C = \frac{1}{2}\left( {\left( {{x^2} + 1} \right)arctgx - x} \right) + C$
Пр.8) $\int {\frac{{\operatorname{arcsinx} dx}}{{{{\sqrt {1 - {x^2}} }^3}}}} = \int {\frac{{\operatorname{arcsinx} dx}}{{\left( {1 - {x^2}} \right)\sqrt {1 - {x^2}} }}} = $
\[\begin{array}{*{20}{c}}
\begin{gathered}
\operatorname{arcsinx} = t \hfill \\
\frac{1}{{\sqrt {1 - {x^2}} }}dx = dt \hfill \\
\end{gathered} & \Rightarrow &\begin{gathered}
x = \sin t \hfill \\
1 - {x^2} = 1 - {\sin ^2}t = {\cos ^2}t \hfill \\
\end{gathered}
\end{array}\]
$ = \int {\frac{{tdt}}{{{{\cos }^2}t}}} = $
\[\begin{array}{*{20}{c}}
\begin{gathered}
u = t \hfill \\
du = dt \hfill \\
\end{gathered} &{}&\begin{gathered}
dv = \frac{{dt}}{{{{\cos }^2}t}} \hfill \\
v = \int {\frac{{dt}}{{{{\cos }^2}t}} = tgt + C} \hfill \\
\end{gathered}
\end{array}\]
$ = t \cdot tgt - \int {tgtdt = } t \cdot tgt - \int {\frac{{\sin t}}{{\cos t}}dt = } $
$\cos t = s$
$ - \sin tdt = ds$
$\sin tdt = - ds$
$ = t \cdot tgt + \int {\frac{{ds}}{s} = } t \cdot tgt + \ln \left| s \right| + C = t \cdot tgt + \ln \left| {\cos t} \right| + C = t \cdot tgt + \ln \left| {\cos t} \right| + C = $
$ = \operatorname{arcsinx} \cdot tg\left( {\operatorname{arcsinx} } \right) + \ln \left| {\cos \left( {\operatorname{arcsinx} } \right)} \right| + C$