Naime imam sljedeći zadatak u bilježnici koji mi nije završen do kraja, a k tome ima mi jedna nejasnoća koju
mi treba pojasnit nakon integracije dobija se sljedeći rezultat pa mi nije jasno kako,zadatak je na slj. slici:
http://yfrog.com/6cdsc001382j
mi treba pojasnit nakon integracije dobija se sljedeći rezultat pa mi nije jasno kako,zadatak je na slj. slici:
http://yfrog.com/6cdsc001382j
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\begin{array}{l}
\int {\sqrt {a^2 - x^2 } dx} = \{ \} \int {\sqrt {a^2 - x^2 } \frac{{\sqrt {a^2 - x^2 } }}{{\sqrt {a^2 - x^2 } }}dx} = \int {\frac{{a^2 - x^2 }}{{\sqrt {a^2 - x^2 } }}dx} = \{ \} \int {\frac{{a^2 }}{{\sqrt {a^2 - x^2 } }}dx} - \int {\frac{{x^2 }}{{\sqrt {a^2 - x^2 } }}dx} = \\
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\int {\frac{{a^2 }}{{\sqrt {a^2 - x^2 } }}dx} = \int {\frac{{a^2 }}{{\sqrt {a^2 - x^2 } }}\frac{{:a}}{{:a}}dx} = \{ \} \int {\frac{{\frac{{a^2 }}{a}}}{{\frac{{\sqrt {a^2 - x^2 } }}{a}}}dx} = \int {\frac{a}{{\sqrt {\frac{{a^2 - x^2 }}{{a^2 }}} }}dx} = \{ \} \int {\frac{a}{{\sqrt {1 - \left( {\frac{x}{a}} \right)^2 } }}dx} = \left| \begin{array}{l}
\frac{x}{a} = t \\
\frac{1}{a}dx = dt \\
dx = adt \\
\end{array} \right| = \int {\frac{a}{{\sqrt {1 - t^2 } }}adt} = \{ \} a^2 \int {\frac{{dt}}{{\sqrt {1 - t^2 } }}} = a^2 \arcsin t = a^2 \arcsin \frac{x}{a} \\
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\int {\frac{{x^2 }}{{\sqrt {a^2 - x^2 } }}dx} = \int {\frac{{x \cdot xdx}}{{\sqrt {a^2 - x^2 } }}} = \{ \} \left| {\begin{array}{*{20}c}
\begin{array}{l}
u = x \\
\\
du = dx \\
\end{array} & \begin{array}{l}
dv = \frac{{xdx}}{{\sqrt {a^2 - x^2 } }} \\
v = \int {\frac{{xdx}}{{\sqrt {a^2 - x^2 } }}} = \left| \begin{array}{l}
a^2 - x^2 = t^2 \\
- 2xdx = 2tdt \\
xdx = - tdt \\
\end{array} \right| = \int {\frac{{ - tdt}}{t}} = - \int {dt} = - t = - \sqrt {a^2 - x^2 } \\
\end{array} \\
\end{array}} \right| = \{ \} u \cdot v - vdu = - x\sqrt {a^2 - x^2 } - \int { - \sqrt {a^2 - x^2 } dx} = \{ \} - x\sqrt {a^2 - x^2 } + \int {\sqrt {a^2 - x^2 } dx} \\
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\int {\sqrt {a^2 - x^2 } dx} = \int {\frac{{a^2 }}{{\sqrt {a^2 - x^2 } }}dx} - \int {\frac{{x^2 }}{{\sqrt {a^2 - x^2 } }}dx} = \{ \} a^2 \arcsin \frac{x}{a} - \left( { - x\sqrt {a^2 - x^2 } + \int {\sqrt {a^2 - x^2 } dx} } \right) = \{ \} a^2 \arcsin \frac{x}{a} + x\sqrt {a^2 - x^2 } - \int {\sqrt {a^2 - x^2 } dx} \\
2\int {\sqrt {a^2 - x^2 } dx} = a^2 \arcsin \frac{x}{a} + x\sqrt {a^2 - x^2 } \\
\int {\sqrt {a^2 - x^2 } dx} = \frac{1}{2}\left( {a^2 \arcsin \frac{x}{a} + x\sqrt {a^2 - x^2 } } \right) = \{ \} \frac{{a^2 }}{2}\arcsin \frac{x}{a} + \frac{x}{2}\sqrt {a^2 - x^2 } \\
\end{array}
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![\[
\int {\sqrt {a^2 - x^2 } dx} = \left[ \begin{array}{l}
x = a\sin t \\
dx = a\cos tdt \\
\end{array} \right] = \int {\sqrt {a^2 - a^2 \sin ^2 t} a\cos tdt} = \int {a^2 \cos ^2 tdt} = a^2 \int {\cos ^2 tdt} = ...
\]](https://static.elitesecurity.org/tex/3d89dc5176ce30e96bb4ee473e43898b.png)
)
