de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral from 0 to 1 of (40)/(x^2-25)

Lösung

\int_{0}^{1} \frac{40}{x ^{2} - 25} d x

Lösung

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

+1

Dezimale

- 1.62186 \dots

Schritte zur Lösung

\int_{0}^{1} \frac{40}{x ^{2} - 25} d x

Entferne die Konstante:

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

= 40 \cdot \int_{0}^{1} \frac{1}{x ^{2} - 25} d x

Wende integrale Substitution an

= 40 \cdot \int_{0}^{\frac{1}{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

= 40 \cdot \frac{1}{5} \cdot \int_{0}^{\frac{1}{5}} \frac{1}{u ^{2} - 1} d u

Faktorisiere

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

= 40 \cdot \frac{1}{5} \cdot \int_{0}^{\frac{1}{5}} \frac{1}{- ( - u ^{2} + 1 )} d u

Entferne die Konstante:

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

= 40 \cdot \frac{1}{5} (- \int_{0}^{\frac{1}{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}

= 40 \cdot \frac{1}{5} (- [\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}]_{0}^{\frac{1}{5}})

Vereinfache

40 \cdot \frac{1}{5} (- [\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}]_{0}^{\frac{1}{5}}) : - 8 [\frac{1}{2} (ln ∣ u + 1 ∣ - ln ∣ u - 1 ∣)]_{0}^{\frac{1}{5}}

= - 8 [\frac{1}{2} (ln ∣ u + 1 ∣ - ln ∣ u - 1 ∣)]_{0}^{\frac{1}{5}}

Berechne die Grenzen:

\frac{ln ( \frac{6}{5} ) - ln ( \frac{4}{5} )}{2}

= - 8 \cdot \frac{ln ( \frac{6}{5} ) - ln ( \frac{4}{5} )}{2}

Vereinfache

= - 4 ln (\frac{3}{2})

Graph

Beliebte Beispiele

integral from 1 to 2 of 2x^3+2x^{-2}

\int_{1}^{2} 2 x^{3} + 2 x^{- 2} d x

integral from 3 to 7 of xe^{-2x}

\int_{3}^{7} x e^{- 2 x} d x

integral from 1 to 3/2 of 3y-2y^2

\int_{1}^{\frac{3}{2}} 3 y - 2 y^{2} d y

integral from 5 to 8 of (30)/x

\int_{5}^{8} \frac{30}{x} d x

integral from 0 to pi of-0.1cos(x)sin(x)

\int_{0}^{π} - 0.1 cos (x) sin (x) d x