de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of (x^2)/(x^2-4)

Lösung

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

Lösung

- ln \frac{x}{2} + 1 + ln \frac{x}{2} - 1 + x + C

Schritte zur Lösung

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

Faktorisiere

x^{2} - 4 : - (- x^{2} + 4)

= \int \frac{x ^{2}}{- ( - x ^{2} + 4 )} d x

Entferne die Konstante:

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

= - \int \frac{x ^{2}}{- x ^{2} + 4} d x

Wende integrale Substitution an

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

Entferne die Konstante:

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

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

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

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

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

\int 1 d u = u

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

Setze in

u = \frac{x}{2}

ein

= - 2 (\frac{ln \frac{x}{2} + 1}{2} - \frac{ln \frac{x}{2} - 1}{2} - \frac{x}{2})

Vereinfache

- 2 (\frac{ln \frac{x}{2} + 1}{2} - \frac{ln \frac{x}{2} - 1}{2} - \frac{x}{2}) : - ln \frac{x}{2} + 1 + ln \frac{x}{2} - 1 + x

= - ln \frac{x}{2} + 1 + ln \frac{x}{2} - 1 + x

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of 13sec^{-2}(xta)n^3x

\int 13 sec^{- 2} (x t a) n^{3} x d x

derivative of ((e^x)/(ln(x)))

\frac{d}{d x} (\frac{( e ^{x} )}{ln ( x )})

tangent cos(x),\at x= pi/2

t an g e n t cos (x), at x = \frac{π}{2}

d/(d{r)}({r}sqrt(1-{r)^2})

\frac{d}{d r} (r 1 - r^{2})

integral of (y+1)/(y^2+y+1)

\int \frac{y + 1}{y ^{2} + y + 1} d y