integral of (x^2)/(1-x^6)

Lösung

\int \frac{x ^{2}}{1 - x ^{6}} d x

\frac{1}{6} (ln x^{3} + 1 - ln x^{3} - 1) + C

Schritte zur Lösung

\int \frac{x ^{2}}{1 - x ^{6}} d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Nutze das gemeinsame Integral :

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

= \frac{1}{3} (\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2})

Setze in

u = x^{3}

ein

= \frac{1}{3} (\frac{ln x ^{3} + 1}{2} - \frac{ln x ^{3} - 1}{2})

Vereinfache

\frac{1}{3} (\frac{ln x ^{3} + 1}{2} - \frac{ln x ^{3} - 1}{2}) : \frac{1}{6} (ln x^{3} + 1 - ln x^{3} - 1)

= \frac{1}{6} (ln x^{3} + 1 - ln x^{3} - 1)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{6} (ln x^{3} + 1 - ln x^{3} - 1) + C

\int x cos (x) d x

x \to - 7 lim (\frac{6 x + 42}{x + 7})

t an g e n t f (x) = 3 x^{3} + 4 x^{2}, (- 1, 1)

\frac{d}{d x} (7^{2 x})

d er i v a t i v e y = \frac{1}{4} x^{2} - \frac{1}{2} ln (x)