integral of (x^7)/((4-x^4)^2)

Lösung

\int \frac{x ^{7}}{( 4 - x ^{4} ) ^{2}} d x

\frac{1}{4} ln 4 - x^{4} + \frac{1}{4 - x ^{4}} + C

Schritte zur Lösung

\int \frac{x ^{7}}{( 4 - x ^{4} ) ^{2}} d x

Wende U-Substitution an

= \int - \frac{- u + 4}{4 u ^{2}} d u

Entferne die Konstante:

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

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

Multipliziere aus

\frac{- u + 4}{u ^{2}} : - \frac{1}{u} + \frac{4}{u ^{2}}

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

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{4}{u ^{2}} d u = - \frac{4}{u}

= - \frac{1}{4} (- ln ∣ u ∣ - \frac{4}{u})

Setze in

u = 4 - x^{4}

ein

= - \frac{1}{4} (- ln 4 - x^{4} - \frac{4}{4 - x ^{4}})

Vereinfache

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

= \frac{1}{4} ln 4 - x^{4} + \frac{1}{4 - x ^{4}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} ln 4 - x^{4} + \frac{1}{4 - x ^{4}} + C

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\int - sin (u) d u