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integral of (sqrt(9-x^2))/(3x)

Lösung

\int \frac{9 - x ^{2}}{3 x} d x

- \frac{1}{2} ln \frac{9 - x ^{2}}{3} + 1 + \frac{1}{2} ln \frac{9 - x ^{2}}{3} - 1 + \frac{1}{3} 9 - x^{2} + C

Schritte zur Lösung

\int \frac{9 - x ^{2}}{3 x} d x

Entferne die Konstante:

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

= \frac{1}{3} \cdot \int \frac{9 - x ^{2}}{x} d x

Wende U-Substitution an

= \frac{1}{3} \cdot \int - \frac{u ^{2}}{- u ^{2} + 9} d u

Entferne die Konstante:

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

= \frac{1}{3} (- \int \frac{u ^{2}}{- u ^{2} + 9} d u)

Wende integrale Substitution an

= \frac{1}{3} (- \int \frac{3 v ^{2}}{- v ^{2} + 1} d v)

Entferne die Konstante:

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

= \frac{1}{3} (- 3 \cdot \int \frac{v ^{2}}{- v ^{2} + 1} d v)

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

= \frac{1}{3} (- 3 \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

= \frac{1}{3} (- 3 (\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

= \frac{1}{3} (- 3 (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2} - v))

Ersetze zurück

= \frac{1}{3} - 3 \frac{ln \frac{9 - x ^{2}}{3} + 1}{2} - \frac{ln \frac{9 - x ^{2}}{3} - 1}{2} - \frac{9 - x ^{2}}{3}

Vereinfache

\frac{1}{3} - 3 \frac{ln \frac{9 - x ^{2}}{3} + 1}{2} - \frac{ln \frac{9 - x ^{2}}{3} - 1}{2} - \frac{9 - x ^{2}}{3} : - \frac{1}{2} ln \frac{9 - x ^{2}}{3} + 1 + \frac{1}{2} ln \frac{9 - x ^{2}}{3} - 1 + \frac{1}{3} 9 - x^{2}

= - \frac{1}{2} ln \frac{9 - x ^{2}}{3} + 1 + \frac{1}{2} ln \frac{9 - x ^{2}}{3} - 1 + \frac{1}{3} 9 - x^{2}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2} ln \frac{9 - x ^{2}}{3} + 1 + \frac{1}{2} ln \frac{9 - x ^{2}}{3} - 1 + \frac{1}{3} 9 - x^{2} + C

p a r i t y sec (sin (x))

\frac{d S}{d t} = 0.9 - \frac{15 S}{1000}

2 x^{3} d y = y (2 x^{2} - y^{3}) d x

\int cos^{3} (t) sin (t) d t

\int \frac{cot ( ln ( x ) )}{x} d x