integral of sqrt(ln(sec(x)))tan(x)

Lösung

\int ln (sec (x)) tan (x) d x

\frac{1}{3 2} ln^{\frac{3}{2}} (sec^{2} (x)) + C

Schritte zur Lösung

\int ln (sec (x)) tan (x) d x

Wende U-Substitution an

= \int \frac{u ln ( 1 + u ^{2} )}{2 ( 1 + u ^{2} )} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int \frac{u ln ( 1 + u ^{2} )}{1 + u ^{2}} d u

Wende U-Substitution an

= \frac{1}{2} \cdot \int v^{2} d v

Wende die Potenzregel an

= \frac{1}{2} \cdot \frac{v ^{3}}{3}

Ersetze zurück

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

Vereinfache

\frac{1}{2} \cdot \frac{( ln ( 1 + tan ^{2} ( x ) ) ) ^{3}}{3} : \frac{1}{3 2} ln^{\frac{3}{2}} (sec^{2} (x))

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

Füge eine Konstante zur Lösung hinzu

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

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