Function Expression :
\[f(x)=x\frac{\sqrt{4-x}}{\sqrt{1+x}} \]Domain
\[\left]-1, 4\right] \]Limits
\[\lim_{x \overset{>}{\rightarrow-1} }f(x) = -\infty \]\[\lim_{x \overset{<}{\rightarrow4} }f(x) = 0 \]
\[ \]
Derivate
\[f^{\,\prime}(x)=- \frac{x \sqrt{4 - x}}{2 \left(x + 1\right)^{\frac{3}{2}}} - \frac{x}{2 \sqrt{4 - x} \sqrt{x + 1}} + \frac{\sqrt{4 - x}}{\sqrt{x + 1}} \]\[f^{\,\prime}(x)=\frac{- x^{2} + \frac{x}{2} + 4}{\sqrt{4 - x} \left(x + 1\right)^{\frac{3}{2}}} \]
\[f^{\,\prime}(x)=\frac{- 2 x^{2} + x + 8}{2 \sqrt{4 - x} \left(x + 1\right)^{\frac{3}{2}}} \]
Integral
\[F(x) = - 8 \left(\begin{cases} - \frac{\sqrt{4 - x} \sqrt{x + 1}}{2} + \frac{5 \operatorname{asin}{\left(\frac{\sqrt{5} \sqrt{4 - x}}{5} \right)}}{2} & \text{for}\: \sqrt{4 - x} > - \sqrt{5} \wedge \sqrt{4 - x} < \sqrt{5} \end{cases}\right) + 2 \left(\begin{cases} \frac{\sqrt{4 - x} \sqrt{x + 1} \cdot \left(2 x - 3\right)}{8} - \frac{5 \sqrt{4 - x} \sqrt{x + 1}}{2} + \frac{75 \operatorname{asin}{\left(\frac{\sqrt{5} \sqrt{4 - x}}{5} \right)}}{8} & \text{for}\: \sqrt{4 - x} > - \sqrt{5} \wedge \sqrt{4 - x} < \sqrt{5} \end{cases}\right) \]Sign Table
Variation Table
Plot
Elapsed Time: 0.0155 seconds