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人気のある微分積分 >

integral of sin^9(x)cos^5(x)

解

\int sin^{9} (x) cos^{5} (x) d x

解

- \frac{cos ^{6} ( x )}{6} + \frac{cos ^{8} ( x )}{2} - \frac{3 cos ^{10} ( x )}{5} + \frac{cos ^{12} ( x )}{3} - \frac{cos ^{14} ( x )}{14} + C

解答ステップ

\int sin^{9} (x) cos^{5} (x) d x

三角関数の公式を使用して書き換える

= \int (1 - cos^{2} (x))^{4} sin (x) cos^{5} (x) d x

u置換積分法を適用する

= \int - u^{5} (1 - u^{2})^{4} d u

拡張

- u^{5} (1 - u^{2})^{4} : - u^{5} + 4 u^{7} - 6 u^{9} + 4 u^{11} - u^{13}

= \int - u^{5} + 4 u^{7} - 6 u^{9} + 4 u^{11} - u^{13} d u

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int u^{5} d u + \int 4 u^{7} d u - \int 6 u^{9} d u + \int 4 u^{11} d u - \int u^{13} d u

\int u^{5} d u = \frac{u ^{6}}{6}

\int 4 u^{7} d u = \frac{u ^{8}}{2}

\int 6 u^{9} d u = \frac{3 u ^{10}}{5}

\int 4 u^{11} d u = \frac{u ^{12}}{3}

\int u^{13} d u = \frac{u ^{14}}{14}

= - \frac{u ^{6}}{6} + \frac{u ^{8}}{2} - \frac{3 u ^{10}}{5} + \frac{u ^{12}}{3} - \frac{u ^{14}}{14}

代用を戻す

u = cos (x)

= - \frac{cos ^{6} ( x )}{6} + \frac{cos ^{8} ( x )}{2} - \frac{3 cos ^{10} ( x )}{5} + \frac{cos ^{12} ( x )}{3} - \frac{cos ^{14} ( x )}{14}

定数を解答に追加する

= - \frac{cos ^{6} ( x )}{6} + \frac{cos ^{8} ( x )}{2} - \frac{3 cos ^{10} ( x )}{5} + \frac{cos ^{12} ( x )}{3} - \frac{cos ^{14} ( x )}{14} + C

グラフ

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