integral (sin^4(x))/(cos(x))

Lösung

integral

\frac{sin ^{4} ( x )}{cos ( x )}

- \frac{sin ^{3} ( x )}{3} - sin (x) + \frac{ln ∣ sin ( x ) + 1 ∣}{2} - \frac{ln ∣ sin ( x ) - 1 ∣}{2} + C

Schritte zur Lösung

\int \frac{sin ^{4} ( x )}{cos ( x )} d x

Trigonometrische Substitution anwenden

= \int \frac{u ^{4}}{1 - u ^{2}} d u

Lange Division

\frac{u ^{4}}{1 - u ^{2}} : - u^{2} - 1 + \frac{1}{- u ^{2} + 1}

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

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

\int 1 d u = u

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

= - \frac{u ^{3}}{3} - u + \frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}

Setze in

u = sin (x)

ein

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

Füge eine Konstante zur Lösung hinzu

= - \frac{sin ^{3} ( x )}{3} - sin (x) + \frac{ln ∣ sin ( x ) + 1 ∣}{2} - \frac{ln ∣ sin ( x ) - 1 ∣}{2} + C

in t e g r a l \frac{1}{( x ^{n} )}

in t e g r a l n^{2}

in t e g r a l 1 + 16 x^{2}

in t e g r a l 2 + x

in t e g r a l \frac{1}{( x ^{\frac{1}{2}} + 1 )}