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Beliebt Rechnen >

integral of (x^2+9)/(x^2+4)

Lösung

\int \frac{x ^{2} + 9}{x ^{2} + 4} d x

Lösung

\frac{5}{2} arctan (\frac{x}{2}) + x + C

Schritte zur Lösung

\int \frac{x ^{2} + 9}{x ^{2} + 4} d x

\frac{x ^{2} + 9}{x ^{2} + 4} = \frac{5}{x ^{2} + 4} + 1

= \int \frac{5}{x ^{2} + 4} + 1 d x

Wende integrale Substitution an

= \int 2 (\frac{5}{4 u ^{2} + 4} + 1) d u

Entferne die Konstante:

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

= 2 \cdot \int \frac{5}{4 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{5}{4 u ^{2} + 4} d u + \int 1 d u)

\int \frac{5}{4 u ^{2} + 4} d u = \frac{5}{4} arctan (u)

\int 1 d u = u

= 2 (\frac{5}{4} arctan (u) + u)

Setze in

u = \frac{x}{2}

ein

= 2 (\frac{5}{4} arctan (\frac{x}{2}) + \frac{x}{2})

Vereinfache

2 (\frac{5}{4} arctan (\frac{x}{2}) + \frac{x}{2}) : \frac{5}{2} arctan (\frac{x}{2}) + x

= \frac{5}{2} arctan (\frac{x}{2}) + x

Füge eine Konstante zur Lösung hinzu

= \frac{5}{2} arctan (\frac{x}{2}) + x + C

Graph

Beliebte Beispiele

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