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^2-4)/(x^2-16)

Lösung

\int \frac{x ^{2} - 4}{x ^{2} - 16} d x

Lösung

- \frac{3}{2} ln \frac{x}{4} + 1 + \frac{3}{2} ln \frac{x}{4} - 1 + x + C

Schritte zur Lösung

\int \frac{x ^{2} - 4}{x ^{2} - 16} d x

Wende integrale Substitution an

= \int \frac{2 ( u ^{2} - 1 )}{u ^{2} - 4} 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} - 1}{u ^{2} - 4} d u

\frac{u ^{2} - 1}{u ^{2} - 4} = \frac{3}{u ^{2} - 4} + 1

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

\int \frac{3}{u ^{2} - 4} d u = - \frac{3}{4} (ln \frac{u}{2} + 1 - ln \frac{u}{2} - 1)

\int 1 d u = u

= 2 (- \frac{3}{4} (ln \frac{u}{2} + 1 - ln \frac{u}{2} - 1) + u)

Setze in

u = \frac{x}{2}

ein

= 2 (- \frac{3}{4} (ln \frac{\frac{x}{2}}{2} + 1 - ln \frac{\frac{x}{2}}{2} - 1) + \frac{x}{2})

Vereinfache

2 (- \frac{3}{4} (ln \frac{\frac{x}{2}}{2} + 1 - ln \frac{\frac{x}{2}}{2} - 1) + \frac{x}{2}) : - \frac{3}{2} ln \frac{x}{4} + 1 + \frac{3}{2} ln \frac{x}{4} - 1 + x

= - \frac{3}{2} ln \frac{x}{4} + 1 + \frac{3}{2} ln \frac{x}{4} - 1 + x

Füge eine Konstante zur Lösung hinzu

= - \frac{3}{2} ln \frac{x}{4} + 1 + \frac{3}{2} ln \frac{x}{4} - 1 + x + C

Graph

Beliebte Beispiele

integral of 1/(x^{0.99)}

\int \frac{1}{x ^{0.99}} d x

integral of (3x^2)/(x^2+x-2)

\int \frac{3 x ^{2}}{x ^{2} + x - 2} d x

integral of (x-2)(4x+2)

\int (x - 2) (4 x + 2) d x

integral of 4.4e^{(-3x+4)}

\int 4.4 e^{(- 3 x + 4)} d x

integral of (2x^3-4x-8)/(x(x-1)(x^2+4))

\int \frac{2 x ^{3} - 4 x - 8}{x ( x - 1 ) ( x ^{2} + 4 )} d x