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人気のある微分積分 >

integral of (sin(x))^3(cos(x))^4

解

\int (sin (x))^{3} (cos (x))^{4} d x

解

- \frac{cos ^{5} ( x )}{5} + \frac{cos ^{7} ( x )}{7} + C

解答ステップ

\int (sin (x))^{3} (cos (x))^{4} d x

簡素化

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

u置換積分法を適用する

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

拡張

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

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

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

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

= - \frac{u ^{5}}{5} + \frac{u ^{7}}{7}

代用を戻す

u = cos (x)

= - \frac{cos ^{5} ( x )}{5} + \frac{cos ^{7} ( x )}{7}

定数を解答に追加する

= - \frac{cos ^{5} ( x )}{5} + \frac{cos ^{7} ( x )}{7} + C

グラフ

人気の例

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