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人気のある微分積分 >

integral of sin^{12}(x)cos^7(x)

解

\int sin^{12} (x) cos^{7} (x) d x

解

\frac{sin ^{13} ( x )}{13} - \frac{sin ^{15} ( x )}{5} + \frac{3 sin ^{17} ( x )}{17} - \frac{sin ^{19} ( x )}{19} + C

解答ステップ

\int sin^{12} (x) cos^{7} (x) d x

三角関数の公式を使用して書き換える

= \int (1 - sin^{2} (x))^{3} cos (x) sin^{12} (x) d x

u置換積分法を適用する

= \int u^{12} (1 - u^{2})^{3} d u

拡張

u^{12} (1 - u^{2})^{3} : u^{12} - 3 u^{14} + 3 u^{16} - u^{18}

= \int u^{12} - 3 u^{14} + 3 u^{16} - u^{18} 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^{12} d u - \int 3 u^{14} d u + \int 3 u^{16} d u - \int u^{18} d u

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

\int 3 u^{14} d u = \frac{u ^{15}}{5}

\int 3 u^{16} d u = \frac{3 u ^{17}}{17}

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

= \frac{u ^{13}}{13} - \frac{u ^{15}}{5} + \frac{3 u ^{17}}{17} - \frac{u ^{19}}{19}

代用を戻す

u = sin (x)

= \frac{sin ^{13} ( x )}{13} - \frac{sin ^{15} ( x )}{5} + \frac{3 sin ^{17} ( x )}{17} - \frac{sin ^{19} ( x )}{19}

定数を解答に追加する

= \frac{sin ^{13} ( x )}{13} - \frac{sin ^{15} ( x )}{5} + \frac{3 sin ^{17} ( x )}{17} - \frac{sin ^{19} ( x )}{19} + C

グラフ

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