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人気のある微分積分 >

integral of sin^7(x)cos^3(x)

解

\int sin^{7} (x) cos^{3} (x) d x

解

- \frac{cos ^{4} ( x )}{4} + \frac{cos ^{6} ( x )}{2} - \frac{3 cos ^{8} ( x )}{8} + \frac{cos ^{10} ( x )}{10} + C

解答ステップ

\int sin^{7} (x) cos^{3} (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))^{3} sin (x) cos^{3} (x) d x

u置換積分法を適用する

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

拡張

- u^{3} (1 - u^{2})^{3} : - u^{3} + 3 u^{5} - 3 u^{7} + u^{9}

= \int - u^{3} + 3 u^{5} - 3 u^{7} + u^{9} 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^{3} d u + \int 3 u^{5} d u - \int 3 u^{7} d u + \int u^{9} d u

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

\int 3 u^{5} d u = \frac{u ^{6}}{2}

\int 3 u^{7} d u = \frac{3 u ^{8}}{8}

\int u^{9} d u = \frac{u ^{10}}{10}

= - \frac{u ^{4}}{4} + \frac{u ^{6}}{2} - \frac{3 u ^{8}}{8} + \frac{u ^{10}}{10}

代用を戻す

u = cos (x)

= - \frac{cos ^{4} ( x )}{4} + \frac{cos ^{6} ( x )}{2} - \frac{3 cos ^{8} ( x )}{8} + \frac{cos ^{10} ( x )}{10}

定数を解答に追加する

= - \frac{cos ^{4} ( x )}{4} + \frac{cos ^{6} ( x )}{2} - \frac{3 cos ^{8} ( x )}{8} + \frac{cos ^{10} ( x )}{10} + C

グラフ

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