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Beliebt Rechnen >

integral of 1/(x^3(x^4+1))

Lösung

\int \frac{1}{x ^{3} ( x ^{4} + 1 )} d x

Lösung

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

Schritte zur Lösung

\int \frac{1}{x ^{3} ( x ^{4} + 1 )} d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Ermittle den Partialbruch von

\frac{1}{u ^{2} ( u ^{2} + 1 )} : \frac{1}{u ^{2}} - \frac{1}{u ^{2} + 1}

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

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

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

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

= \frac{1}{2} (- \frac{1}{u} - arctan (u))

Setze in

u = x^{2}

ein

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of sqrt(1+4x^2(x^2-1))

\int 1 + 4 x^{2} (x^{2} - 1) d x

integral of (e^{6x}+2)/(e^{6x)+e^{-6x}}

\int \frac{e ^{6 x} + 2}{e ^{6 x} + e ^{- 6 x}} d x

integral of e^8

\int e^{8} d x

integral of (5x-4)/(x^2+7x+6)

\int \frac{5 x - 4}{x ^{2} + 7 x + 6} d x

integral x^{3/2}

in t e g r a l x^{\frac{3}{2}}