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人気のある微分積分 >

integral of cot^4(x)csc^4(x)

解

\int cot^{4} (x) csc^{4} (x) d x

解

- \frac{1}{7} cot^{7} (x) - \frac{1}{5} cot^{5} (x) + C

解答ステップ

\int cot^{4} (x) csc^{4} (x) d x

三角関数による置換を適用する

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

定数を除く:

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

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

拡張

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

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

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

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

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

代用を戻す

u = cot (x)

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

簡素化

- (\frac{cot ^{7} ( x )}{7} + \frac{cot ^{5} ( x )}{5}) : - \frac{1}{7} cot^{7} (x) - \frac{1}{5} cot^{5} (x)

= - \frac{1}{7} cot^{7} (x) - \frac{1}{5} cot^{5} (x)

定数を解答に追加する

= - \frac{1}{7} cot^{7} (x) - \frac{1}{5} cot^{5} (x) + C

グラフ

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