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Keith_Richards [23]
3 years ago
10

Explain your reasoning as you compare the values of a^n and a^-n when n<0 brainly

Mathematics
1 answer:
mojhsa [17]3 years ago
4 0

n < 0, is another way to say "n is negative", so let's check


\bf ~~~~~~~~~~~~\textit{negative exponents}&#10;\\\\&#10;a^{-n} \implies \cfrac{1}{a^n}&#10;\qquad \qquad&#10;\cfrac{1}{a^n}\implies a^{-n}&#10;\qquad \qquad&#10;a^n\implies \cfrac{1}{a^{-n}}&#10;\\\\[-0.35em]&#10;\rule{34em}{0.25pt}\\\\&#10;a^n~\hspace{10.5em}\stackrel{n = -n}{a^{-n}}\implies \cfrac{1}{a^n}&#10;\\\\\\&#10;a^{-n}~\hspace{10em}\stackrel{n=-n}{a^{-(-n)}}\implies a^{+n}\implies a^n

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Answer:

The solution is  \frac{1}{10} * tan^{-1}[\frac{e^{2x}}{5} ] +  C

Step-by-step explanation:

From the question

    The function given is  f(x) =  \frac{e^{2x}}{ 25 + e^{4x}} dx

The  indefinite integral is  mathematically represented as

          \int\limits  {\frac{e^{2x}}{ 25 + e^{4x}}} \, dx

Now  let  e^{2x} =  u

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=>   2 e^{2x}dx =  du

So

\int\limits  {\frac{e^{2x}}{ 25 + e^{4x}}} \, dx =  \int\limits  {\frac{1}{ 2(25 + u^2)} } \, du

= \frac{1}{2} \int\limits  {\frac{1}{ 25 + u^2)} } \, du

=  \frac{1}{2} \int\limits  {\frac{1}{ 5^2 + u^2)} } \, du

= \frac{1}{2} \frac{tan^{-1} [\frac{u}{5} ]}{5}  +  C

Now substituting for  u

\frac{1}{10} * tan^{-1}[\frac{e^{2x}}{5} ] +  C

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