- #1
HotMintea
- 43
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1. The problem statement
Use dimensional analysis to find [itex] \int\sqrt{\ a\ - \ b\ x^2\ }\ dx [/itex].
A useful result is [itex] \int\sqrt{\ 1\ - \ x^2\ }\ dx\ = \frac{arcsin{x}}{2}\ + \frac{x\sqrt{\ 1\ - \ x^2\ }}{2}\ + \ C [/itex].
2. The attempt at a solution
If I let [itex] = L^2 [/itex] and [itex] [x] = M [/itex], then [itex] [a] = L^2 M^2 [/itex] and [itex] [\int\sqrt{\ a\ - \ b\ x^2\ }\ dx]\ = LM^2 [/itex].
Hence, my answer was:
[tex]
\begin{equation*}
\begin{split}
\int\sqrt{\ a\ - \ b\ x^2\ }\ dx\ = \frac{a}{\sqrt{b}}\frac{arcsin{\frac{\sqrt{b}\ x}{\sqrt{a}}}}{2}\ + \frac{x\sqrt{\ a\ - \ b\ x^2\ }}{2}\ + \ C.
\end{split}
\end{equation*}
[/tex]
However, the correct answer (by Wolfram Alpha) was:
[tex]
\begin{equation*}
\begin{split}
\int\sqrt{\ a\ - \ b\ x^2\ }\ dx\ = \frac{a}{\sqrt{b}}\frac{arctan{\frac{\sqrt{b}\ x}{\sqrt{a\ - \ bx^2\ }}}}{2}\ + \frac{x\sqrt{\ a\ - \ b\ x^2\ }}{2}\ + \ C.
\end{split}
\end{equation*}
[/tex]
( http://www.wolframalpha.com/input/?i=int+sqrt%28a-bx^2%29dx [/URL])
I wonder why [itex] \int\sqrt{\ a\ - \ b\ x^2\ }\ dx\ [/itex] will not be the same as [itex] \int\sqrt{\ 1\ - \ x^2\ }\ dx\ [/itex] when a = b = 1. Moreover, I would like to know how to find [itex] arctan{\frac{\sqrt{b}\ x}{\sqrt{a\ - \ bx^2\ }}} [/itex] part by dimensional analysis or similar method without doing the full integral.
Use dimensional analysis to find [itex] \int\sqrt{\ a\ - \ b\ x^2\ }\ dx [/itex].
A useful result is [itex] \int\sqrt{\ 1\ - \ x^2\ }\ dx\ = \frac{arcsin{x}}{2}\ + \frac{x\sqrt{\ 1\ - \ x^2\ }}{2}\ + \ C [/itex].
2. The attempt at a solution
If I let [itex] = L^2 [/itex] and [itex] [x] = M [/itex], then [itex] [a] = L^2 M^2 [/itex] and [itex] [\int\sqrt{\ a\ - \ b\ x^2\ }\ dx]\ = LM^2 [/itex].
Hence, my answer was:
[tex]
\begin{equation*}
\begin{split}
\int\sqrt{\ a\ - \ b\ x^2\ }\ dx\ = \frac{a}{\sqrt{b}}\frac{arcsin{\frac{\sqrt{b}\ x}{\sqrt{a}}}}{2}\ + \frac{x\sqrt{\ a\ - \ b\ x^2\ }}{2}\ + \ C.
\end{split}
\end{equation*}
[/tex]
However, the correct answer (by Wolfram Alpha) was:
[tex]
\begin{equation*}
\begin{split}
\int\sqrt{\ a\ - \ b\ x^2\ }\ dx\ = \frac{a}{\sqrt{b}}\frac{arctan{\frac{\sqrt{b}\ x}{\sqrt{a\ - \ bx^2\ }}}}{2}\ + \frac{x\sqrt{\ a\ - \ b\ x^2\ }}{2}\ + \ C.
\end{split}
\end{equation*}
[/tex]
( http://www.wolframalpha.com/input/?i=int+sqrt%28a-bx^2%29dx [/URL])
I wonder why [itex] \int\sqrt{\ a\ - \ b\ x^2\ }\ dx\ [/itex] will not be the same as [itex] \int\sqrt{\ 1\ - \ x^2\ }\ dx\ [/itex] when a = b = 1. Moreover, I would like to know how to find [itex] arctan{\frac{\sqrt{b}\ x}{\sqrt{a\ - \ bx^2\ }}} [/itex] part by dimensional analysis or similar method without doing the full integral.
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