integral of (sqrt(1-x^4))/(x^5)

Lösung

\int \frac{1 - x ^{4}}{x ^{5}} d x

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

Schritte zur Lösung

\int \frac{1 - x ^{4}}{x ^{5}} d x

Wende U-Substitution an

= \int \frac{1 - u ^{2}}{2 u ^{3}} 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 ^{3}} d u

Wende die partielle Integration an

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

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

= \frac{1}{2} (- \frac{1 - u ^{2}}{2 u ^{2}} - (- \frac{1}{4} (ln - u^{2} + 1 + 1 - ln - u^{2} + 1 - 1)))

Setze in

u = x^{2}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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