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Beliebt Rechnen >

integral of 1/(x^4+4)

Lösung

\int \frac{1}{x ^{4} + 4} d x

Lösung

\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)) + C

Schritte zur Lösung

\int \frac{1}{x ^{4} + 4} d x

Ermittle den Partialbruch von

\frac{1}{x ^{4} + 4} : \frac{x + 2}{8 ( x ^{2} + 2 x + 2 )} + \frac{- x + 2}{8 ( x ^{2} - 2 x + 2 )}

= \int \frac{x + 2}{8 ( x ^{2} + 2 x + 2 )} + \frac{- x + 2}{8 ( x ^{2} - 2 x + 2 )} d x

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

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

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

Füge eine Konstante zur Lösung hinzu

= \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)) + C

Graph

Beliebte Beispiele

derivative y=(x^2-2sqrt(x))/x

d er i v a t i v e y = \frac{x ^{2} - 2 x}{x}

(\partial)/(\partial v)(2v)

\frac{\partial}{\partial v} (2 v)

implicit (dy)/(dx),y=-4x^5

im pl i c i t \frac{d y}{d x}, y = - 4 x^{5}

derivative y=(sqrt(x)-4)/(sqrt(x)+4)

d er i v a t i v e y = \frac{x - 4}{x + 4}

integral of 10+6x+24x^2

\int 10 + 6 x + 24 x^{2} d x