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

Lösung

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

\frac{x sin ^{4} ( x )}{cos ^{4} ( x )} - x tan^{4} (x) + \frac{1}{3} tan^{3} (x) + x - tan (x) + C

Schritte zur Lösung

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

Wende die partielle Integration an

= \frac{x sin ^{4} ( x )}{cos ^{4} ( x )} - \int 4 x tan^{3} (x) sec^{2} (x) d x

\int 4 x tan^{3} (x) sec^{2} (x) d x = x tan^{4} (x) - \frac{1}{3} tan^{3} (x) - x + tan (x)

= \frac{x sin ^{4} ( x )}{cos ^{4} ( x )} - (x tan^{4} (x) - \frac{1}{3} tan^{3} (x) - x + tan (x))

Vereinfache

= \frac{x sin ^{4} ( x )}{cos ^{4} ( x )} - x tan^{4} (x) + \frac{1}{3} tan^{3} (x) + x - tan (x)

Füge eine Konstante zur Lösung hinzu

= \frac{x sin ^{4} ( x )}{cos ^{4} ( x )} - x tan^{4} (x) + \frac{1}{3} tan^{3} (x) + x - tan (x) + C

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