Function Expression :
\[f(x)=\frac{\sqrt{x-1}}{\sqrt{(x+3 )(x-5 )}} \]Domain
\[\left[5, \infty\right[ \]Limits
\[\lim_{x \overset{>}{\rightarrow5} }f(x) = +\infty \]\[\lim_{x \rightarrow+\infty}f(x) = 0 \]
\[ \]
Derivate
\[f^{\,\prime}(x)=\frac{1}{2 \sqrt{\left(x - 5\right) \left(x + 3\right)} \sqrt{x - 1}} - \frac{\sqrt{x - 1} \left(x - 1\right)}{\sqrt{\left(x - 5\right) \left(x + 3\right)} \left(x - 5\right) \left(x + 3\right)} \]\[f^{\,\prime}(x)=\frac{\sqrt{\left(x - 5\right) \left(x + 3\right)} \left(\frac{\left(x - 5\right) \left(x + 3\right)}{2} - \left(x - 1\right)^{2}\right)}{\left(x - 5\right)^{2} \sqrt{x - 1} \left(x + 3\right)^{2}} \]
\[f^{\,\prime}(x)=\frac{\sqrt{\left(x - 5\right) \left(x + 3\right)} \left(\left(x - 5\right) \left(x + 3\right) - 2 \left(x - 1\right)^{2}\right)}{2 \left(x - 5\right)^{2} \sqrt{x - 1} \left(x + 3\right)^{2}} \]
Integral
\[F(x) = \int \frac{\sqrt{x - 1}}{\sqrt{\left(x - 5\right) \left(x + 3\right)}}\, dx \]Sign Table
Variation Table
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