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人気のある微分積分 >

integral of cot^3(ax)csc^3(ax)

解

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

解

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

解答ステップ

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

u置換積分法を適用する

= \int \frac{cot ^{3} ( u ) csc ^{3} ( u )}{a} d u

定数を除く:

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

= \frac{1}{a} \cdot \int cot^{3} (u) csc^{3} (u) d u

u置換積分法を適用する

= \frac{1}{a} \cdot \int - v^{2} (v^{2} - 1) d v

定数を除く:

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

= \frac{1}{a} (- \int v^{2} (v^{2} - 1) d v)

拡張

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

= \frac{1}{a} (- \int v^{4} - v^{2} d v)

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

= \frac{1}{a} (- (\int v^{4} d v - \int v^{2} d v))

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

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

= \frac{1}{a} (- (\frac{v ^{5}}{5} - \frac{v ^{3}}{3}))

代用を戻す

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

簡素化

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

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

定数を解答に追加する

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

グラフ

人気の例

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\frac{d}{d x} ((6 x^{3} + 5 x^{2})^{2})

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x \to - 2 lim (\frac{5 x}{2 + x})

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\int \frac{x}{a} d x

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n = 1 \sum \infty n (n - 1)

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t an g e n t f (x) = 3^{x}, at x = 8