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Beliebt Rechnen >

integral of 15tan^2(x)sec^3(x)

Lösung

\int 15 tan^{2} (x) sec^{3} (x) d x

Lösung

15 (- \frac{1}{4} (\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣) + \frac{1}{4} sec^{4} (x) sin (x)) + C

Schritte zur Lösung

\int 15 tan^{2} (x) sec^{3} (x) d x

Entferne die Konstante:

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

= 15 \cdot \int tan^{2} (x) sec^{3} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 15 \cdot \int (- 1 + sec^{2} (x)) sec^{3} (x) d x

Multipliziere aus

(- 1 + sec^{2} (x)) sec^{3} (x) : - sec^{3} (x) + sec^{5} (x)

= 15 \cdot \int - sec^{3} (x) + sec^{5} (x) d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 15 (- \int sec^{3} (x) d x + \int sec^{5} (x) d x)

\int sec^{3} (x) d x = \frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣

\int sec^{5} (x) d x = \frac{sec ^{4} ( x ) sin ( x )}{4} + \frac{3}{4} (\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣)

= 15 (- (\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣) + \frac{sec ^{4} ( x ) sin ( x )}{4} + \frac{3}{4} (\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣))

Vereinfache

15 (- (\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣) + \frac{sec ^{4} ( x ) sin ( x )}{4} + \frac{3}{4} (\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣)) : 15 (- \frac{1}{4} (\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣) + \frac{1}{4} sec^{4} (x) sin (x))

= 15 (- \frac{1}{4} (\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣) + \frac{1}{4} sec^{4} (x) sin (x))

Füge eine Konstante zur Lösung hinzu

= 15 (- \frac{1}{4} (\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣) + \frac{1}{4} sec^{4} (x) sin (x)) + C

Graph

Beliebte Beispiele

derivative 2xe^{4x}

d er i v a t i v e 2 x e^{4 x}

integral of 2/(x^3sqrt(x^2-3))

\int \frac{2}{x ^{3} x ^{2} - 3} d x

derivative sqrt(2)x+sqrt(3x)

d er i v a t i v e 2 x + 3 x

y^{''}-3y^'+2y=e^{3x}

y^{^{''}} - 3 y^{^{'}} + 2 y = e^{3 x}

limit as x approaches infinity of 1/x+5

x \to \infty lim (\frac{1}{x} + 5)