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人気のある微分積分 >

integral of sin^4(x)cos^9(x)

解

\int sin^{4} (x) cos^{9} (x) d x

解

\frac{sin ^{5} ( x )}{5} - \frac{4 sin ^{7} ( x )}{7} + \frac{2 sin ^{9} ( x )}{3} - \frac{4 sin ^{11} ( x )}{11} + \frac{sin ^{13} ( x )}{13} + C

解答ステップ

\int sin^{4} (x) cos^{9} (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))^{4} cos (x) sin^{4} (x) d x

u置換積分法を適用する

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

拡張

u^{4} (1 - u^{2})^{4} : u^{4} - 4 u^{6} + 6 u^{8} - 4 u^{10} + u^{12}

= \int u^{4} - 4 u^{6} + 6 u^{8} - 4 u^{10} + u^{12} 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^{4} d u - \int 4 u^{6} d u + \int 6 u^{8} d u - \int 4 u^{10} d u + \int u^{12} d u

\int u^{4} d u = \frac{u ^{5}}{5}

\int 4 u^{6} d u = \frac{4 u ^{7}}{7}

\int 6 u^{8} d u = \frac{2 u ^{9}}{3}

\int 4 u^{10} d u = \frac{4 u ^{11}}{11}

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

= \frac{u ^{5}}{5} - \frac{4 u ^{7}}{7} + \frac{2 u ^{9}}{3} - \frac{4 u ^{11}}{11} + \frac{u ^{13}}{13}

代用を戻す

u = sin (x)

= \frac{sin ^{5} ( x )}{5} - \frac{4 sin ^{7} ( x )}{7} + \frac{2 sin ^{9} ( x )}{3} - \frac{4 sin ^{11} ( x )}{11} + \frac{sin ^{13} ( x )}{13}

定数を解答に追加する

= \frac{sin ^{5} ( x )}{5} - \frac{4 sin ^{7} ( x )}{7} + \frac{2 sin ^{9} ( x )}{3} - \frac{4 sin ^{11} ( x )}{11} + \frac{sin ^{13} ( x )}{13} + C

グラフ

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