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integral of (sqrt(7-x^4))/(5x)

Lösung

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

- \frac{1}{20} (7 ln \frac{- x ^{4} + 7}{7} + 1 - 7 ln \frac{- x ^{4} + 7}{7} - 1 - 2 - x^{4} + 7) + C

Schritte zur Lösung

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

Entferne die Konstante:

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

= \frac{1}{5} \cdot \int \frac{7 - x ^{4}}{x} d x

Wende U-Substitution an

= \frac{1}{5} \cdot \int \frac{7 - u}{4 u} d u

Entferne die Konstante:

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

= \frac{1}{5} \cdot \frac{1}{4} \cdot \int \frac{7 - u}{u} d u

Wende U-Substitution an

= \frac{1}{5} \cdot \frac{1}{4} \cdot \int - \frac{2 v ^{2}}{- v ^{2} + 7} d v

Entferne die Konstante:

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

= \frac{1}{5} \cdot \frac{1}{4} (- 2 \cdot \int \frac{v ^{2}}{- v ^{2} + 7} d v)

Wende integrale Substitution an

= \frac{1}{5} \cdot \frac{1}{4} (- 2 \cdot \int \frac{7 w ^{2}}{- w ^{2} + 1} d w)

Entferne die Konstante:

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

= \frac{1}{5} \cdot \frac{1}{4} (- 27 \cdot \int \frac{w ^{2}}{- w ^{2} + 1} d w)

\frac{w ^{2}}{- w ^{2} + 1} = \frac{1}{- w ^{2} + 1} - 1

= \frac{1}{5} \cdot \frac{1}{4} (- 27 \cdot \int \frac{1}{- w ^{2} + 1} - 1 d w)

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}{5} \cdot \frac{1}{4} (- 27 (\int \frac{1}{- w ^{2} + 1} d w - \int 1 d w))

\int \frac{1}{- w ^{2} + 1} d w = \frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}

\int 1 d w = w

= \frac{1}{5} \cdot \frac{1}{4} (- 27 (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2} - w))

Ersetze zurück

= \frac{1}{5} \cdot \frac{1}{4} - 27 \frac{ln \frac{7 - x ^{4}}{7} + 1}{2} - \frac{ln \frac{7 - x ^{4}}{7} - 1}{2} - \frac{7 - x ^{4}}{7}

Vereinfache

\frac{1}{5} \cdot \frac{1}{4} - 27 \frac{ln \frac{7 - x ^{4}}{7} + 1}{2} - \frac{ln \frac{7 - x ^{4}}{7} - 1}{2} - \frac{7 - x ^{4}}{7} : - \frac{1}{20} (7 ln \frac{- x ^{4} + 7}{7} + 1 - 7 ln \frac{- x ^{4} + 7}{7} - 1 - 2 - x^{4} + 7)

= - \frac{1}{20} (7 ln \frac{- x ^{4} + 7}{7} + 1 - 7 ln \frac{- x ^{4} + 7}{7} - 1 - 2 - x^{4} + 7)

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{20} (7 ln \frac{- x ^{4} + 7}{7} + 1 - 7 ln \frac{- x ^{4} + 7}{7} - 1 - 2 - x^{4} + 7) + C

d er i v a t i v e 7 x - 2 y

x \to - 3 lim (\frac{x ^{2} + 3}{4})

x \to 1 lim (e)

n = 1 \sum \infty (\frac{1}{3})^{k - n}

d er i v a t i v e 5 sec^{2} (x)