integral of sin^3(x)(2-3cos(x))^2

Lösung

\int sin^{3} (x) (2 - 3 cos (x))^{2} d x

4 (- cos (x) + \frac{cos ^{3} ( x )}{3}) - 3 sin^{4} (x) + 9 (- \frac{cos ^{3} ( x )}{3} + \frac{cos ^{5} ( x )}{5}) + C

Schritte zur Lösung

\int sin^{3} (x) (2 - 3 cos (x))^{2} d x

Multipliziere aus

sin^{3} (x) (2 - 3 cos (x))^{2} : 4 sin^{3} (x) - 12 sin^{3} (x) cos (x) + 9 sin^{3} (x) cos^{2} (x)

= \int 4 sin^{3} (x) - 12 sin^{3} (x) cos (x) + 9 sin^{3} (x) cos^{2} (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 4 sin^{3} (x) d x - \int 12 sin^{3} (x) cos (x) d x + \int 9 sin^{3} (x) cos^{2} (x) d x

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

\int 12 sin^{3} (x) cos (x) d x = 3 sin^{4} (x)

\int 9 sin^{3} (x) cos^{2} (x) d x = 9 (- \frac{cos ^{3} ( x )}{3} + \frac{cos ^{5} ( x )}{5})

= 4 (- cos (x) + \frac{cos ^{3} ( x )}{3}) - 3 sin^{4} (x) + 9 (- \frac{cos ^{3} ( x )}{3} + \frac{cos ^{5} ( x )}{5})

Füge eine Konstante zur Lösung hinzu

= 4 (- cos (x) + \frac{cos ^{3} ( x )}{3}) - 3 sin^{4} (x) + 9 (- \frac{cos ^{3} ( x )}{3} + \frac{cos ^{5} ( x )}{5}) + C

\int (x^{7} + 1) d x

\int ln (e^{(2 x - 1)}) d x

\int \frac{x ^{\frac{1}{2}}}{2} d x

\int (e^{- \frac{t}{6}} - e^{\frac{t}{6}})^{2} d t

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