integral of (5sqrt(ln^2(x)+9))/x

Lösung

\int \frac{5 ln ^{2} ( x ) + 9}{x} d x

45 (\frac{1}{18} ln (x) 9 + ln^{2} (x) + \frac{1}{2} ln (\frac{1}{3} ln (x) + 9 + ln^{2} (x))) + C

Schritte zur Lösung

\int \frac{5 ln ^{2} ( x ) + 9}{x} d x

Entferne die Konstante:

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

= 5 \cdot \int \frac{ln ^{2} ( x ) + 9}{x} d x

Wende U-Substitution an

= 5 \cdot \int u^{2} + 9 d u

Trigonometrische Substitution anwenden

= 5 \cdot \int 9 sec^{3} (v) d v

Entferne die Konstante:

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

= 5 \cdot 9 \cdot \int sec^{3} (v) d v

Wende integrale Reduktion an

= 5 \cdot 9 (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} \cdot \int sec (v) d v)

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

= 5 \cdot 9 (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

Ersetze zurück

= 5 \cdot 9 (\frac{sec ^{2} ( arctan ( \frac{1}{3} ln ( x ) ) ) sin ( arctan ( \frac{1}{3} ln ( x ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{3} ln (x))) + sec (arctan (\frac{1}{3} ln (x))))

Vereinfache

5 \cdot 9 (\frac{sec ^{2} ( arctan ( \frac{1}{3} ln ( x ) ) ) sin ( arctan ( \frac{1}{3} ln ( x ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{3} ln (x))) + sec (arctan (\frac{1}{3} ln (x)))) : 45 (\frac{1}{18} ln (x) 9 + ln^{2} (x) + \frac{1}{2} ln (\frac{1}{3} ln (x) + 9 + ln^{2} (x)))

= 45 (\frac{1}{18} ln (x) 9 + ln^{2} (x) + \frac{1}{2} ln (\frac{1}{3} ln (x) + 9 + ln^{2} (x)))

Füge eine Konstante zur Lösung hinzu

= 45 (\frac{1}{18} ln (x) 9 + ln^{2} (x) + \frac{1}{2} ln (\frac{1}{3} ln (x) + 9 + ln^{2} (x))) + C

\frac{\partial}{\partial x} (y^{2} + x y + 3 y + 2 x + 3)

\int \frac{1}{5 sin ( x ) + 3} d x

\frac{\partial}{\partial x} (x^{2} + y^{2} + 3 x y + 15 x)

\frac{\partial}{\partial t} (e^{- t} sin (t))

\int (a^{5 x} + e^{5 x}) d x