integral of ((5+ln(x))^5)/x

Lösung

\int \frac{( 5 + ln ( x ) ) ^{5}}{x} d x

3125 ln ∣ x ∣ + \frac{3125}{2} ln^{2} (x) + \frac{1250}{3} ln^{3} (x) + \frac{125}{2} ln^{4} (x) + 5 ln^{5} (x) + \frac{ln ^{6} ( x )}{6} + C

Schritte zur Lösung

\int \frac{( 5 + ln ( x ) ) ^{5}}{x} d x

Multipliziere aus

\frac{( 5 + ln ( x ) ) ^{5}}{x} : \frac{3125}{x} + \frac{3125 ln ( x )}{x} + \frac{1250 ln ^{2} ( x )}{x} + \frac{250 ln ^{3} ( x )}{x} + \frac{25 ln ^{4} ( x )}{x} + \frac{ln ^{5} ( x )}{x}

= \int \frac{3125}{x} + \frac{3125 ln ( x )}{x} + \frac{1250 ln ^{2} ( x )}{x} + \frac{250 ln ^{3} ( x )}{x} + \frac{25 ln ^{4} ( x )}{x} + \frac{ln ^{5} ( x )}{x} d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{3125}{x} d x + \int \frac{3125 ln ( x )}{x} d x + \int \frac{1250 ln ^{2} ( x )}{x} d x + \int \frac{250 ln ^{3} ( x )}{x} d x + \int \frac{25 ln ^{4} ( x )}{x} d x + \int \frac{ln ^{5} ( x )}{x} d x

\int \frac{3125}{x} d x = 3125 ln ∣ x ∣

\int \frac{3125 ln ( x )}{x} d x = \frac{3125}{2} ln^{2} (x)

\int \frac{1250 ln ^{2} ( x )}{x} d x = \frac{1250}{3} ln^{3} (x)

\int \frac{250 ln ^{3} ( x )}{x} d x = \frac{125}{2} ln^{4} (x)

\int \frac{25 ln ^{4} ( x )}{x} d x = 5 ln^{5} (x)

\int \frac{ln ^{5} ( x )}{x} d x = \frac{ln ^{6} ( x )}{6}

= 3125 ln ∣ x ∣ + \frac{3125}{2} ln^{2} (x) + \frac{1250}{3} ln^{3} (x) + \frac{125}{2} ln^{4} (x) + 5 ln^{5} (x) + \frac{ln ^{6} ( x )}{6}

Füge eine Konstante zur Lösung hinzu

= 3125 ln ∣ x ∣ + \frac{3125}{2} ln^{2} (x) + \frac{1250}{3} ln^{3} (x) + \frac{125}{2} ln^{4} (x) + 5 ln^{5} (x) + \frac{ln ^{6} ( x )}{6} + C

d er i v a t i v e f (x) = 6 cos^{4} (3 x + 2)

x \to 5 lim (x^{3} - x^{2} + 5 x)

y^{^{'}} = y tan (t) + sin (t)

x \to 1 lim (x^{5} - 2 x - 6)

\frac{d}{d x} (\frac{csc ( x )}{1 + csc ( x )})