integral from 3 to 4 of 1/(4-x^2)

Lösung

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

\frac{ln ( 3 ) - ln ( 5 )}{4}

Dezimale

- 0.12770 \dots

Schritte zur Lösung

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

Wende integrale Substitution an

= \int_{\frac{3}{2}}^{2} \frac{1}{2 ( - u ^{2} + 1 )} 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{3}{2}}^{2} \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}

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

Vereinfache

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

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

Berechne die Grenzen:

\frac{ln ( 3 ) - ln ( \frac{5}{2} ) - ln ( 2 )}{2}

= \frac{1}{2} \cdot \frac{ln ( 3 ) - ln ( \frac{5}{2} ) - ln ( 2 )}{2}

Vereinfache

= \frac{ln ( 3 ) - ln ( 5 )}{4}

\int_{- 4}^{5} (3 x^{2} - 12 + x^{3}) d x

\int_{4}^{5} \frac{4}{x ^{2} - 1} d x

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\int_{1 - y}^{- y^{2} + y + 1} 2 y d y

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