integral of (4sqrt(ln^2(x)+49))/x

Lösung

\int \frac{4 ln ^{2} ( x ) + 49}{x} d x

196 (\frac{1}{98} ln (x) 49 + ln^{2} (x) + \frac{1}{2} ln (\frac{1}{7} ln (x) + 49 + ln^{2} (x))) + C

Schritte zur Lösung

\int \frac{4 ln ^{2} ( x ) + 49}{x} d x

Entferne die Konstante:

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

= 4 \cdot \int \frac{ln ^{2} ( x ) + 49}{x} d x

Wende U-Substitution an

= 4 \cdot \int u^{2} + 49 d u

Trigonometrische Substitution anwenden

= 4 \cdot \int 49 sec^{3} (v) d v

Entferne die Konstante:

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

= 4 \cdot 49 \cdot \int sec^{3} (v) d v

Wende integrale Reduktion an

= 4 \cdot 49 (\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) ∣

= 4 \cdot 49 (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

Ersetze zurück

= 4 \cdot 49 (\frac{sec ^{2} ( arctan ( \frac{1}{7} ln ( x ) ) ) sin ( arctan ( \frac{1}{7} ln ( x ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{7} ln (x))) + sec (arctan (\frac{1}{7} ln (x))))

Vereinfache

4 \cdot 49 (\frac{sec ^{2} ( arctan ( \frac{1}{7} ln ( x ) ) ) sin ( arctan ( \frac{1}{7} ln ( x ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{7} ln (x))) + sec (arctan (\frac{1}{7} ln (x)))) : 196 (\frac{1}{98} ln (x) 49 + ln^{2} (x) + \frac{1}{2} ln (\frac{1}{7} ln (x) + 49 + ln^{2} (x)))

= 196 (\frac{1}{98} ln (x) 49 + ln^{2} (x) + \frac{1}{2} ln (\frac{1}{7} ln (x) + 49 + ln^{2} (x)))

Füge eine Konstante zur Lösung hinzu

= 196 (\frac{1}{98} ln (x) 49 + ln^{2} (x) + \frac{1}{2} ln (\frac{1}{7} ln (x) + 49 + ln^{2} (x))) + C

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

\frac{d}{d x} (\frac{7 e ^{x} + 4 e ^{- x}}{8})

\frac{d}{d x} (ln (\frac{( x - 4 )}{x - 3}))

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