integral of cos^{12}(x)sin^{15}(x)

Lösung

\int cos^{12} (x) sin^{15} (x) d x

- \frac{cos ^{13} ( x )}{13} + \frac{7 cos ^{15} ( x )}{15} - \frac{21 cos ^{17} ( x )}{17} + \frac{35 cos ^{19} ( x )}{19} - \frac{5 cos ^{21} ( x )}{3} + \frac{21 cos ^{23} ( x )}{23} - \frac{7 cos ^{25} ( x )}{25} + \frac{cos ^{27} ( x )}{27} + C

Schritte zur Lösung

\int cos^{12} (x) sin^{15} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int (1 - cos^{2} (x))^{7} sin (x) cos^{12} (x) d x

Wende U-Substitution an

= \int - u^{12} (1 - u^{2})^{7} d u

Multipliziere aus

- u^{12} (1 - u^{2})^{7} : - u^{12} + 7 u^{14} - 21 u^{16} + 35 u^{18} - 35 u^{20} + 21 u^{22} - 7 u^{24} + u^{26}

= \int - u^{12} + 7 u^{14} - 21 u^{16} + 35 u^{18} - 35 u^{20} + 21 u^{22} - 7 u^{24} + u^{26} 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^{12} d u + \int 7 u^{14} d u - \int 21 u^{16} d u + \int 35 u^{18} d u - \int 35 u^{20} d u + \int 21 u^{22} d u - \int 7 u^{24} d u + \int u^{26} d u

\int u^{12} d u = \frac{u ^{13}}{13}

\int 7 u^{14} d u = \frac{7 u ^{15}}{15}

\int 21 u^{16} d u = \frac{21 u ^{17}}{17}

\int 35 u^{18} d u = \frac{35 u ^{19}}{19}

\int 35 u^{20} d u = \frac{5 u ^{21}}{3}

\int 21 u^{22} d u = \frac{21 u ^{23}}{23}

\int 7 u^{24} d u = \frac{7 u ^{25}}{25}

\int u^{26} d u = \frac{u ^{27}}{27}

= - \frac{u ^{13}}{13} + \frac{7 u ^{15}}{15} - \frac{21 u ^{17}}{17} + \frac{35 u ^{19}}{19} - \frac{5 u ^{21}}{3} + \frac{21 u ^{23}}{23} - \frac{7 u ^{25}}{25} + \frac{u ^{27}}{27}

Setze in

u = cos (x)

ein

= - \frac{cos ^{13} ( x )}{13} + \frac{7 cos ^{15} ( x )}{15} - \frac{21 cos ^{17} ( x )}{17} + \frac{35 cos ^{19} ( x )}{19} - \frac{5 cos ^{21} ( x )}{3} + \frac{21 cos ^{23} ( x )}{23} - \frac{7 cos ^{25} ( x )}{25} + \frac{cos ^{27} ( x )}{27}

Füge eine Konstante zur Lösung hinzu

= - \frac{cos ^{13} ( x )}{13} + \frac{7 cos ^{15} ( x )}{15} - \frac{21 cos ^{17} ( x )}{17} + \frac{35 cos ^{19} ( x )}{19} - \frac{5 cos ^{21} ( x )}{3} + \frac{21 cos ^{23} ( x )}{23} - \frac{7 cos ^{25} ( x )}{25} + \frac{cos ^{27} ( x )}{27} + C

l a pl a ce t r an s f or m cos (15 t)

\frac{d}{d x} (\frac{2 x}{2 + x ^{2}})

y^{^{''}} + 5 y^{^{'}} + 6 y = 0, y (0) = 2, y^{^{'}} (0) = 0

in v erse l a pl a ce \frac{\frac{1}{s + 1} - 1}{s ^{2} - 1}

\int z^{2} (\frac{1}{2 π} e^{- \frac{1}{2} z^{2}}) d z