Неодређени интеграли – парцијална интеграција 4

Текст задатака објашњених у видео лекцији:

Пр.9)     Решити $\int {{e^x}\sin xdx} $

Пр.10)   $\int {\cos \left( {\ln x} \right)} dx$

Пр.9)     $\int {{e^x}\sin xdx}= $

\[\begin{array}{*{20}{c}}
\begin{gathered}
u = \sin x \hfill \\
du = \cos x \hfill \\
\end{gathered} &{}&\begin{gathered}
dv = {e^x}dx \hfill \\
v = \int {{e^x}} dx = {e^x} + C \hfill \\
\end{gathered}
\end{array}\]

$ = {e^x}\sin x - \int {{e^x}\cos xdx = } $

\[\begin{array}{*{20}{c}}
\begin{gathered}
\cos x = u \hfill \\
- \sin xdx = du \hfill \\
\end{gathered} &{}&\begin{gathered}
dv = {e^x}dx \hfill \\
v = {e^x} + C \hfill \\
\end{gathered}
\end{array}\]

$ = {e^x}\sin x - \left( {{e^x}\cos x + \int {{e^x}\sin xdx} } \right) = {e^x}\sin x - {e^x}\cos x - \int {{e^x}\sin xdx} $

$\int {{e^x}\sin xdx}  = {e^x}\sin x - {e^x}\cos x - \int {{e^x}\sin xdx} $

$2\int {{e^x}\sin xdx}  = {e^x}\left( {\sin x - \cos x} \right)$

$\int {{e^x}\sin xdx}  = \frac{{{e^x}}}{2}\left( {\sin x - \cos x} \right)$

Пр.10)   $\int {\cos \left( {\ln x} \right)} dx=$

\[\begin{array}{*{20}{c}}
\begin{gathered}
u = \cos \left( {\ln x} \right) \hfill \\
du = - \sin \left( {\ln x} \right) \cdot {\left( {\ln x} \right)^\prime }dx \hfill \\
du = - \frac{{\sin \left( {\ln x} \right)}}{x}dx \hfill \\
\end{gathered} &{}&\begin{gathered}
dv = dx \hfill \\
v = x + C \hfill \\
\end{gathered}
\end{array}\]

$ = x \cdot \cos \left( {\ln x} \right) + \int {x\frac{{\sin \left( {\ln x} \right)}}{x}dx = } $

\[\begin{array}{*{20}{c}}
\begin{gathered}
u = \sin \left( {\ln x} \right) \hfill \\
du = \frac{{\cos \left( {\ln x} \right)}}{x}dx \hfill \\
\end{gathered} &{}&\begin{gathered}
dv = dx \hfill \\
v = x + C \hfill \\
\end{gathered}
\end{array}\]

$= x \cdot \cos \left( {\ln x} \right) + x\sin \left( {\ln x} \right) - \int {x\frac{{\cos \left( {\ln x} \right)}}{x}dx} $

$\int {\cos \left( {\ln x} \right)} dx = x \cdot \cos \left( {\ln x} \right) + x\sin \left( {\ln x} \right) - \int {\cos \left( {\ln x} \right)dx} $

$2\int {\cos \left( {\ln x} \right)} dx = x \cdot \cos \left( {\ln x} \right) + x\sin \left( {\ln x} \right)$

$\int {\cos \left( {\ln x} \right)} dx = \frac{x}{2} \cdot \left( {\cos \left( {\ln x} \right) + \sin \left( {\ln x} \right)} \right)$