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Beliebt Rechnen >

integral of 1/(cos^3(x)sin(x))

Lösung

\int \frac{1}{cos ^{3} ( x ) sin ( x )} d x

Lösung

\frac{tan ^{2} ( x )}{2} + ln ∣ tan (x) ∣ + C

Schritte zur Lösung

\int \frac{1}{cos ^{3} ( x ) sin ( x )} d x

Trigonometrische Substitution anwenden

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

Vereinfache

(u^{2} + 1)^{\frac{1}{2}} u^{2} + 1 : u^{2} + 1

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

Multipliziere aus

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

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

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

\int \frac{1}{u} d u = ln ∣ u ∣

= \frac{u ^{2}}{2} + ln ∣ u ∣

Setze in

u = tan (x)

ein

= \frac{tan ^{2} ( x )}{2} + ln ∣ tan (x) ∣

Füge eine Konstante zur Lösung hinzu

= \frac{tan ^{2} ( x )}{2} + ln ∣ tan (x) ∣ + C

Graph

Beliebte Beispiele

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