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人気のある微分積分 >

integral of 8cot^3(x)csc^3(x)

解

\int 8 cot^{3} (x) csc^{3} (x) d x

解

- 8 (\frac{1}{5} csc^{5} (x) - \frac{1}{3} csc^{3} (x)) + C

解答ステップ

\int 8 cot^{3} (x) csc^{3} (x) d x

定数を除く:

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

= 8 \cdot \int cot^{3} (x) csc^{3} (x) d x

u置換積分法を適用する

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

定数を除く:

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

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

拡張

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

= 8 (- \int u^{4} - u^{2} 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

= 8 (- (\int u^{4} d u - \int u^{2} d u))

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

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

= 8 (- (\frac{u ^{5}}{5} - \frac{u ^{3}}{3}))

代用を戻す

u = csc (x)

= 8 (- (\frac{csc ^{5} ( x )}{5} - \frac{csc ^{3} ( x )}{3}))

簡素化

8 (- (\frac{csc ^{5} ( x )}{5} - \frac{csc ^{3} ( x )}{3})) : - 8 (\frac{1}{5} csc^{5} (x) - \frac{1}{3} csc^{3} (x))

= - 8 (\frac{1}{5} csc^{5} (x) - \frac{1}{3} csc^{3} (x))

定数を解答に追加する

= - 8 (\frac{1}{5} csc^{5} (x) - \frac{1}{3} csc^{3} (x)) + C

グラフ

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