integral of sec^{-1/2}(x)tan^3(x)

Lösung

\int sec^{- \frac{1}{2}} (x) tan^{3} (x) d x

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

Schritte zur Lösung

\int sec^{- \frac{1}{2}} (x) tan^{3} (x) d x

Wende U-Substitution an

= \int \frac{u ^{2} - 1}{u u} d u

Multipliziere aus

\frac{u ^{2} - 1}{u u} : u - \frac{1}{u u}

= \int u - \frac{1}{u 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

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

\int u d u = \frac{2}{3} u^{\frac{3}{2}}

\int \frac{1}{u u} d u = - \frac{2}{u}

= \frac{2}{3} u^{\frac{3}{2}} - (- \frac{2}{u})

Setze in

u = sec (x)

ein

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

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