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Beliebt Rechnen >

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

Lösung

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

Lösung

arctan (\frac{x}{2}) + \frac{1}{2} ln \frac{x ^{2}}{4} + 1 + C

Schritte zur Lösung

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

Wende integrale Substitution an

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

Multipliziere aus

\frac{1 + u}{u ^{2} + 1} : \frac{1}{u ^{2} + 1} + \frac{u}{u ^{2} + 1}

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

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

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

= arctan (u) + \frac{1}{2} ln u^{2} + 1

Setze in

u = \frac{x}{2}

ein

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

(\frac{x}{2})^{2} = \frac{x ^{2}}{4}

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

Füge eine Konstante zur Lösung hinzu

= arctan (\frac{x}{2}) + \frac{1}{2} ln \frac{x ^{2}}{4} + 1 + C

Graph

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