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人気のある微分積分 >

integral of sin^5(x)*cos^8(x)

解

\int sin^{5} (x) \cdot cos^{8} (x) d x

解

- \frac{cos ^{9} ( x )}{9} + \frac{2 cos ^{11} ( x )}{11} - \frac{cos ^{13} ( x )}{13} + C

解答ステップ

\int sin^{5} (x) cos^{8} (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))^{2} sin (x) cos^{8} (x) d x

u置換積分法を適用する

= \int - u^{8} (1 - u^{2})^{2} d u

拡張

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

= \int - u^{8} + 2 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^{8} d u + \int 2 u^{10} d u - \int u^{12} d u

\int u^{8} d u = \frac{u ^{9}}{9}

\int 2 u^{10} d u = \frac{2 u ^{11}}{11}

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

= - \frac{u ^{9}}{9} + \frac{2 u ^{11}}{11} - \frac{u ^{13}}{13}

代用を戻す

u = cos (x)

= - \frac{cos ^{9} ( x )}{9} + \frac{2 cos ^{11} ( x )}{11} - \frac{cos ^{13} ( x )}{13}

定数を解答に追加する

= - \frac{cos ^{9} ( x )}{9} + \frac{2 cos ^{11} ( x )}{11} - \frac{cos ^{13} ( x )}{13} + C

グラフ

人気の例

integral of (5x^{3/2}-3)

\int (5 x^{\frac{3}{2}} - 3) d x

integral of e^{10x}5x

\int e^{10 x} 5 x d x

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\int 3 x ln (6 x) d x

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