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integral of 1/(xsqrt(25-4x^2))

Lösung

\int \frac{1}{x 25 - 4 x ^{2}} d x

- \frac{1}{10} (ln \frac{25 - 4 x ^{2}}{5} + 1 - ln \frac{25 - 4 x ^{2}}{5} - 1) + C

Schritte zur Lösung

\int \frac{1}{x 25 - 4 x ^{2}} d x

Wende U-Substitution an

= \int \frac{1}{2 u 25 - 4 u} 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 25 - 4 u} d u

Wende U-Substitution an

= \frac{1}{2} \cdot \int \frac{2}{v ^{2} - 25} d v

Entferne die Konstante:

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

= \frac{1}{2} \cdot 2 \cdot \int \frac{1}{v ^{2} - 25} d v

Wende integrale Substitution an

= \frac{1}{2} \cdot 2 \cdot \int \frac{1}{5 ( w ^{2} - 1 )} d w

Entferne die Konstante:

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

= \frac{1}{2} \cdot 2 \cdot \frac{1}{5} \cdot \int \frac{1}{w ^{2} - 1} d w

Faktorisiere

w^{2} - 1 : - (- w^{2} + 1)

= \frac{1}{2} \cdot 2 \cdot \frac{1}{5} \cdot \int \frac{1}{- ( - w ^{2} + 1 )} d w

Entferne die Konstante:

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

= \frac{1}{2} \cdot 2 \cdot \frac{1}{5} (- \int \frac{1}{- w ^{2} + 1} d w)

Nutze das gemeinsame Integral :

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

= \frac{1}{2} \cdot 2 \cdot \frac{1}{5} (- (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}))

Ersetze zurück

= \frac{1}{2} \cdot 2 \cdot \frac{1}{5} - \frac{ln \frac{25 - 4 x ^{2}}{5} + 1}{2} - \frac{ln \frac{25 - 4 x ^{2}}{5} - 1}{2}

Vereinfache

\frac{1}{2} \cdot 2 \cdot \frac{1}{5} - \frac{ln \frac{25 - 4 x ^{2}}{5} + 1}{2} - \frac{ln \frac{25 - 4 x ^{2}}{5} - 1}{2} : - \frac{1}{10} (ln \frac{25 - 4 x ^{2}}{5} + 1 - ln \frac{25 - 4 x ^{2}}{5} - 1)

= - \frac{1}{10} (ln \frac{25 - 4 x ^{2}}{5} + 1 - ln \frac{25 - 4 x ^{2}}{5} - 1)

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{10} (ln \frac{25 - 4 x ^{2}}{5} + 1 - ln \frac{25 - 4 x ^{2}}{5} - 1) + C

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