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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{1}{2} arctan (x + 1) + \frac{1}{2} arctan (x - 1) + C

Schritte zur Lösung

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

Multipliziere aus

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

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

\int \frac{2}{x ^{4} + 4} d x = 2 (\frac{1}{16} (ln x^{2} + 2 x + 2 + 2 arctan (x + 1)) + \frac{1}{16} (- ln x^{2} - 2 x + 2 + 2 arctan (x - 1)))

= - \frac{1}{8} (ln x^{2} + 2 x + 2 - 2 arctan (x + 1)) + \frac{1}{8} (ln x^{2} - 2 x + 2 + 2 arctan (x - 1)) + 2 (\frac{1}{16} (ln x^{2} + 2 x + 2 + 2 arctan (x + 1)) + \frac{1}{16} (- ln x^{2} - 2 x + 2 + 2 arctan (x - 1)))

Vereinfache

- \frac{1}{8} (ln x^{2} + 2 x + 2 - 2 arctan (x + 1)) + \frac{1}{8} (ln x^{2} - 2 x + 2 + 2 arctan (x - 1)) + 2 (\frac{1}{16} (ln x^{2} + 2 x + 2 + 2 arctan (x + 1)) + \frac{1}{16} (- ln x^{2} - 2 x + 2 + 2 arctan (x - 1))) : \frac{1}{2} arctan (x + 1) + \frac{1}{2} arctan (x - 1)

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

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} arctan (x + 1) + \frac{1}{2} arctan (x - 1) + C

Graph

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