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人気のある微分積分 >

integral of 10sin^6(x)cos^3(x)

解

\int 10 sin^{6} (x) cos^{3} (x) d x

解

10 (\frac{sin ^{7} ( x )}{7} - \frac{sin ^{9} ( x )}{9}) + C

解答ステップ

\int 10 sin^{6} (x) cos^{3} (x) d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 10 \cdot \int sin^{6} (x) cos^{3} (x) d x

三角関数の公式を使用して書き換える

= 10 \cdot \int (1 - sin^{2} (x)) cos (x) sin^{6} (x) d x

u置換積分法を適用する

= 10 \cdot \int u^{6} (1 - u^{2}) d u

拡張

u^{6} (1 - u^{2}) : u^{6} - u^{8}

= 10 \cdot \int u^{6} - u^{8} 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

= 10 (\int u^{6} d u - \int u^{8} d u)

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

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

= 10 (\frac{u ^{7}}{7} - \frac{u ^{9}}{9})

代用を戻す

u = sin (x)

= 10 (\frac{sin ^{7} ( x )}{7} - \frac{sin ^{9} ( x )}{9})

定数を解答に追加する

= 10 (\frac{sin ^{7} ( x )}{7} - \frac{sin ^{9} ( x )}{9}) + C

グラフ

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