de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of (2x^2+2x-7)/(x^2-4)

Lösung

\int \frac{2 x ^{2} + 2 x - 7}{x ^{2} - 4} d x

Lösung

2 x - \frac{1}{4} ln \frac{x}{2} + 1 + \frac{1}{4} ln \frac{x}{2} - 1 + ln x^{2} - 4 + C

Schritte zur Lösung

\int \frac{2 x ^{2} + 2 x - 7}{x ^{2} - 4} d x

Multipliziere aus

\frac{2 x ^{2} + 2 x - 7}{x ^{2} - 4} : \frac{2 x ^{2}}{x ^{2} - 4} + \frac{2 x}{x ^{2} - 4} - \frac{7}{x ^{2} - 4}

Wende die Summenregel an:

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

= \int \frac{2 x ^{2}}{x ^{2} - 4} d x + \int \frac{2 x}{x ^{2} - 4} d x - \int \frac{7}{x ^{2} - 4} d x

\int \frac{2 x ^{2}}{x ^{2} - 4} d x = - 4 (\frac{1}{2} ln \frac{x}{2} + 1 - \frac{1}{2} ln \frac{x}{2} - 1 - \frac{x}{2})

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

\int \frac{7}{x ^{2} - 4} d x = - \frac{7}{4} (ln \frac{x}{2} + 1 - ln \frac{x}{2} - 1)

= - 4 (\frac{1}{2} ln \frac{x}{2} + 1 - \frac{1}{2} ln \frac{x}{2} - 1 - \frac{x}{2}) + ln x^{2} - 4 - (- \frac{7}{4} (ln \frac{x}{2} + 1 - ln \frac{x}{2} - 1))

Vereinfache

- 4 (\frac{1}{2} ln \frac{x}{2} + 1 - \frac{1}{2} ln \frac{x}{2} - 1 - \frac{x}{2}) + ln x^{2} - 4 - (- \frac{7}{4} (ln \frac{x}{2} + 1 - ln \frac{x}{2} - 1)) : 2 x - \frac{1}{4} ln \frac{x}{2} + 1 + \frac{1}{4} ln \frac{x}{2} - 1 + ln x^{2} - 4

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

Füge eine Konstante zur Lösung hinzu

= 2 x - \frac{1}{4} ln \frac{x}{2} + 1 + \frac{1}{4} ln \frac{x}{2} - 1 + ln x^{2} - 4 + C

Graph

Beliebte Beispiele

integral of 7x^{2/5}-18x^{4/5}

\int 7 x^{\frac{2}{5}} - 18 x^{\frac{4}{5}} d x

integral of-(x^2)/(x^2+1)

\int - \frac{x ^{2}}{x ^{2} + 1} d x

integral of (5-x^2)

\int (5 - x^{2}) d x

integral of (x+sin(x))/(cos(x)+1)

\int \frac{x + sin ( x )}{cos ( x ) + 1} d x

integral of 4/(3\sqrt[3]{x)}

\int \frac{4}{3 3 x} d x