integral of 1/(x(x^7+1))

Lösung

\int \frac{1}{x ( x ^{7} + 1 )} d x

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

Schritte zur Lösung

\int \frac{1}{x ( x ^{7} + 1 )} d x

Wende U-Substitution an

= \int \frac{1}{7 ( u - 1 ) u} d u

Entferne die Konstante:

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

= \frac{1}{7} \cdot \int \frac{1}{( u - 1 ) u} d u

Ermittle den Partialbruch von

\frac{1}{( u - 1 ) u} : \frac{1}{u - 1} - \frac{1}{u}

= \frac{1}{7} \cdot \int \frac{1}{u - 1} - \frac{1}{u} 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}{7} (\int \frac{1}{u - 1} d u - \int \frac{1}{u} d u)

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

\int \frac{1}{u} d u = ln ∣ u ∣

= \frac{1}{7} (ln ∣ u - 1 ∣ - ln ∣ u ∣)

Setze in

u = 1 + x^{7}

ein

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

Vereinfache

\frac{1}{7} (ln 1 + x^{7} - 1 - ln 1 + x^{7}) : \frac{1}{7} (ln x^{7} - ln 1 + x^{7})

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{125}{( 5 x - 300 ) ^{2}} d x

\int \frac{3}{1 + 9 x ^{2}} d x

\int x sin (3 x - 1) d x

\int \frac{tan ( x )}{ln ( cos ( x ) )} d x

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