+26367 Hallo! hoffe es kennt sich jemand aus: \(\begin{array}{rcl} \int \sqrt[5]{ x(\frac{1}{x}-5 )}\ dx \end{array}\)
\(\begin{array}{rcl} \int \sqrt[5]{ x(\frac{1}{x}-5 )}\ dx \\ &=& \int \sqrt[5]{ (1-5x )}\ dx \\ &=& \int (1-5x )^\frac15\ dx \\\\ \qquad \text{ Substitution } z&=& 1-5x\\ dz &=& -5\ dx\\ dx &=& -\frac15 \ dz\\\\ \int \sqrt[5]{ x(\frac{1}{x}-5 )}\ dx &=& \int z^{\frac15}\cdot ( -\frac15 \ dz ) \\ &=& -\frac15 \int z^{\frac15}\ dz \quad \boxed{~ \text{Formel } \int x^n \ dx = \frac{x^{n+1}}{n+1} ~}\\ &=& -\frac15\left( \frac{ z^{\frac15 + 1} } { \frac15 + 1 } \right)+c\\ &=& -\frac15\left( \frac{ z^{\frac65} } { \frac65 } \right)+c\\ &=& -\frac15\cdot \frac56 z^{\frac65}+c\\ &=& -\frac16 z^{\frac65}+c \qquad z= 1-5x\\ &=& -\frac16 (1-5x)^{\frac65}+c\\ \mathbf{ \int \sqrt[5]{ x(\frac{1}{x}-5 )}\ dx } & \mathbf{=} & \mathbf{-\frac16 (1-5x)^{\frac65} +c} \end{array}\)
+26367 Hallo! hoffe es kennt sich jemand aus: \(\begin{array}{rcl} \int \sqrt[5]{ x(\frac{1}{x}-5 )}\ dx \end{array}\)
\(\begin{array}{rcl} \int \sqrt[5]{ x(\frac{1}{x}-5 )}\ dx \\ &=& \int \sqrt[5]{ (1-5x )}\ dx \\ &=& \int (1-5x )^\frac15\ dx \\\\ \qquad \text{ Substitution } z&=& 1-5x\\ dz &=& -5\ dx\\ dx &=& -\frac15 \ dz\\\\ \int \sqrt[5]{ x(\frac{1}{x}-5 )}\ dx &=& \int z^{\frac15}\cdot ( -\frac15 \ dz ) \\ &=& -\frac15 \int z^{\frac15}\ dz \quad \boxed{~ \text{Formel } \int x^n \ dx = \frac{x^{n+1}}{n+1} ~}\\ &=& -\frac15\left( \frac{ z^{\frac15 + 1} } { \frac15 + 1 } \right)+c\\ &=& -\frac15\left( \frac{ z^{\frac65} } { \frac65 } \right)+c\\ &=& -\frac15\cdot \frac56 z^{\frac65}+c\\ &=& -\frac16 z^{\frac65}+c \qquad z= 1-5x\\ &=& -\frac16 (1-5x)^{\frac65}+c\\ \mathbf{ \int \sqrt[5]{ x(\frac{1}{x}-5 )}\ dx } & \mathbf{=} & \mathbf{-\frac16 (1-5x)^{\frac65} +c} \end{array}\)
