integral of (sin^5(5x)+cos^3(5x))

Lösung

\int (sin^{5} (5 x) + cos^{3} (5 x)) d x

\frac{1}{5} (- cos (5 x) + \frac{2 cos ^{3} ( 5 x )}{3} - \frac{cos ^{5} ( 5 x )}{5}) + \frac{1}{5} (sin (5 x) - \frac{sin ^{3} ( 5 x )}{3}) + C

Schritte zur Lösung

\int sin^{5} (5 x) + cos^{3} (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

= \int sin^{5} (5 x) d x + \int cos^{3} (5 x) d x

\int sin^{5} (5 x) d x = \frac{1}{5} (- cos (5 x) + \frac{2 cos ^{3} ( 5 x )}{3} - \frac{cos ^{5} ( 5 x )}{5})

\int cos^{3} (5 x) d x = \frac{1}{5} (sin (5 x) - \frac{sin ^{3} ( 5 x )}{3})

= \frac{1}{5} (- cos (5 x) + \frac{2 cos ^{3} ( 5 x )}{3} - \frac{cos ^{5} ( 5 x )}{5}) + \frac{1}{5} (sin (5 x) - \frac{sin ^{3} ( 5 x )}{3})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{5} (- cos (5 x) + \frac{2 cos ^{3} ( 5 x )}{3} - \frac{cos ^{5} ( 5 x )}{5}) + \frac{1}{5} (sin (5 x) - \frac{sin ^{3} ( 5 x )}{3}) + C

\int (\frac{7}{5})^{x} d x

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