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Beliebt Rechnen >

integral of (x+4)/((x+2)sqrt(x+5))

Lösung

\int \frac{x + 4}{( x + 2 ) x + 5} d x

Lösung

2 (- \frac{1}{3} (ln \frac{1}{3} (x + 5) + 1 - ln \frac{1}{3} (x + 5) - 1) + x + 5) + C

Schritte zur Lösung

\int \frac{x + 4}{( x + 2 ) x + 5} d x

Wende U-Substitution an

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

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

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

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

\int 1 d u = u

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

Setze in

u = x + 5

ein

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

Vereinfache

2 (- \frac{1}{3} (ln \frac{x + 5}{3} + 1 - ln \frac{x + 5}{3} - 1) + x + 5) : 2 (- \frac{1}{3} (ln \frac{1}{3} (x + 5) + 1 - ln \frac{1}{3} (x + 5) - 1) + x + 5)

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

Füge eine Konstante zur Lösung hinzu

= 2 (- \frac{1}{3} (ln \frac{1}{3} (x + 5) + 1 - ln \frac{1}{3} (x + 5) - 1) + x + 5) + C

Graph

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