integral from 3 to 4 of 2/(25-x^2)

Lösung

\int_{3}^{4} \frac{2}{25 - x ^{2}} d x

\frac{ln ( \frac{9}{4} )}{5}

Dezimale

0.16218 \dots

Schritte zur Lösung

\int_{3}^{4} \frac{2}{25 - x ^{2}} d x

Entferne die Konstante:

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

= 2 \cdot \int_{3}^{4} \frac{1}{25 - x ^{2}} d x

Wende integrale Substitution an

= 2 \cdot \int_{\frac{3}{5}}^{\frac{4}{5}} \frac{1}{5 ( - u ^{2} + 1 )} d u

Entferne die Konstante:

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

= 2 \cdot \frac{1}{5} \cdot \int_{\frac{3}{5}}^{\frac{4}{5}} \frac{1}{- u ^{2} + 1} d u

Nutze das gemeinsame Integral :

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

= 2 \cdot \frac{1}{5} [\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}]_{\frac{3}{5}}^{\frac{4}{5}}

Vereinfache

2 \cdot \frac{1}{5} [\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}]_{\frac{3}{5}}^{\frac{4}{5}} : \frac{2}{5} [\frac{ln ∣ u + 1 ∣ - ln ∣ u - 1 ∣}{2}]_{\frac{3}{5}}^{\frac{4}{5}}

= \frac{2}{5} [\frac{ln ∣ u + 1 ∣ - ln ∣ u - 1 ∣}{2}]_{\frac{3}{5}}^{\frac{4}{5}}

Berechne die Grenzen:

\frac{ln ( \frac{9}{5} ) + ln ( 5 ) - ln ( \frac{8}{5} ) + ln ( \frac{2}{5} )}{2}

= \frac{2}{5} \cdot \frac{ln ( \frac{9}{5} ) + ln ( 5 ) - ln ( \frac{8}{5} ) + ln ( \frac{2}{5} )}{2}

Vereinfache

= \frac{ln ( \frac{9}{4} )}{5}

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