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Homework Statement
I want to prove the standard result for indefinite integrals that [tex] \int \frac{1}{a^2-x^2} .dx\ =\ \frac{1}{2a}\ln{(\frac{a+x}{a-x})} \\ \mbox{But the problem is that i am unable to do so. I get this--}
\int {\frac{1}{a^2-x^2} .dx } \\ [/tex]
[tex]
= \frac{1}{2a}\int {\frac{2a}{a^2-x^2} .dx} \\
= \frac{1}{2a}\int{ \frac{a+x+a-x}{a^2-x^2}} \\ [/tex]
[tex]
= \frac{1}{2a}\int {\frac{1}{a-x}+\frac{1}{a+x} .dx} \\
[/tex]
[tex]
= \frac{1}{2a}(\int {\frac{1}{a-x}}+\int{\frac{1}{a+x}} ) \\
[/tex]
[tex]
= \frac{1}{2a}\ln{(a-x)}+ln{(a+x)} .dx [/tex]
I used this from an earlier derivation of [tex]
\int{\frac{1}{x^2-a^2} } \mbox{where } \frac{1}{2a}(\int{ \frac{1}{x-a}}+\int{\frac{1}{x+a}} ) [/tex]
was converted to [tex]
= \frac{1}{2a}\ln{(x-a)}-ln{(x-a)} .dx [/tex] similarly Plz suggest where i went wrong and suggest a way to get the desired proof
Thx
EDIT: There's some trouble with latex. The code i entered was -: (space to avoid repetition)
[i tex] \int \frac{1}{a^2-x^2} .dx\ =\ \frac{1}{2a}\ln{(\frac{a+x}{a-x})} \\ \mbox{But the problem is that i am unable to do so. I get this--}
\int {\frac{1}{a^2-x^2} .dx } \\
= \frac{1}{2a}\int {\frac{2a}{a^2-x^2} .dx} \\
= \frac{1}{2a}\int{ \frac{a+x+a-x}{a^2-x^2}} \\
= \frac{1}{2a}\int {\frac{1}{a-x}+\frac{1}{a+x} .dx} \\
= \frac{1}{2a}(\int {\frac{1}{a-x}}+\int{\frac{1}{a+x}} ) \\
= \frac{1}{2a}\ln{a-x}+ln{a+x} .dx mbox{I used this from an earlier derivation of }
\int{\frac{1}{x^2-a^2} } \ mbox{where } \frac{1}{2a}(\int{ \frac{1}{x-a}}+\int{\frac{1}{x+a}} )
mbox{ was converted to ln similarly } [/i tex]
I am unable to get the complete msg?
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