integral of (sqrt(49-x^2))/x

Lösung

\int \frac{49 - x ^{2}}{x} d x

- \frac{7}{2} ln \frac{49 - x ^{2}}{7} + 1 + \frac{7}{2} ln \frac{49 - x ^{2}}{7} - 1 + 49 - x^{2} + C

Schritte zur Lösung

\int \frac{49 - x ^{2}}{x} d x

Wende U-Substitution an

= \int - \frac{u ^{2}}{- u ^{2} + 49} d u

Entferne die Konstante:

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

= - \int \frac{u ^{2}}{- u ^{2} + 49} d u

Wende integrale Substitution an

= - \int \frac{7 v ^{2}}{- v ^{2} + 1} d v

Entferne die Konstante:

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

= - 7 \cdot \int \frac{v ^{2}}{- v ^{2} + 1} d v

\frac{v ^{2}}{- v ^{2} + 1} = \frac{1}{- v ^{2} + 1} - 1

= - 7 \cdot \int \frac{1}{- v ^{2} + 1} - 1 d v

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - 7 (\int \frac{1}{- v ^{2} + 1} d v - \int 1 d v)

\int \frac{1}{- v ^{2} + 1} d v = \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}

\int 1 d v = v

= - 7 (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2} - v)

Ersetze zurück

= - 7 \frac{ln \frac{49 - x ^{2}}{7} + 1}{2} - \frac{ln \frac{49 - x ^{2}}{7} - 1}{2} - \frac{49 - x ^{2}}{7}

Vereinfache

- 7 \frac{ln \frac{49 - x ^{2}}{7} + 1}{2} - \frac{ln \frac{49 - x ^{2}}{7} - 1}{2} - \frac{49 - x ^{2}}{7} : - \frac{7}{2} ln \frac{49 - x ^{2}}{7} + 1 + \frac{7}{2} ln \frac{49 - x ^{2}}{7} - 1 + 49 - x^{2}

= - \frac{7}{2} ln \frac{49 - x ^{2}}{7} + 1 + \frac{7}{2} ln \frac{49 - x ^{2}}{7} - 1 + 49 - x^{2}

Füge eine Konstante zur Lösung hinzu

= - \frac{7}{2} ln \frac{49 - x ^{2}}{7} + 1 + \frac{7}{2} ln \frac{49 - x ^{2}}{7} - 1 + 49 - x^{2} + C

d er i v a t i v e (0.4 x + 2) (0.5 x - 5)

\frac{d}{d x} (x^{7} e^{10 x})

\frac{d}{d x} ((x - 2)^{4})

y^{^{''}} - 30 y^{^{'}} + 225 y = 0

y \to \frac{1}{2} lim ((16 y^{2} - 4)^{4})