integral of r/((r^2+x^2)^{3/2)}

Lösung

\int \frac{r}{( r ^{2} + x ^{2} ) ^{\frac{3}{2}}} d r

- \frac{1}{( x ^{2} + r ^{2} ) ^{\frac{1}{2}}} + C

Schritte zur Lösung

\int \frac{r}{( r ^{2} + x ^{2} ) ^{\frac{3}{2}}} d r

Wende U-Substitution an

= \int \frac{1}{2 u ^{\frac{3}{2}}} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{1}{u ^{\frac{3}{2}}} d u

Wende die Potenzregel an

= \frac{1}{2} (- \frac{2}{u ^{\frac{1}{2}}})

Setze in

u = x^{2} + r^{2}

ein

= \frac{1}{2} (- \frac{2}{( x ^{2} + r ^{2} ) ^{\frac{1}{2}}})

Vereinfache

\frac{1}{2} (- \frac{2}{( x ^{2} + r ^{2} ) ^{\frac{1}{2}}}) : - \frac{1}{( x ^{2} + r ^{2} ) ^{\frac{1}{2}}}

= - \frac{1}{( x ^{2} + r ^{2} ) ^{\frac{1}{2}}}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{( x ^{2} + r ^{2} ) ^{\frac{1}{2}}} + C

x \to 6 - lim (\frac{5}{x - 6})

\frac{\partial}{\partial y} (1 - x^{2})

x \to 0 + lim (x^{17} ln (x))

\int_{3}^{12} ln (x^{8}) d x

y^{^{'}} = - \frac{1}{2} y