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人気のある微分積分 >

integral of cot^5(6x)

解

\int cot^{5} (6 x) d x

解

- \frac{1}{6} (- \frac{1}{2} cot^{2} (6 x) + \frac{1}{4} cot^{4} (6 x) + \frac{1}{2} ln (csc^{2} (6 x))) + C

解答ステップ

\int cot^{5} (6 x) d x

u置換積分法を適用する

= \int cot^{5} (u) \frac{1}{6} d u

定数を除く:

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

= \frac{1}{6} \cdot \int cot^{5} (u) d u

u置換積分法を適用する

= \frac{1}{6} \cdot \int - \frac{v ^{5}}{1 + v ^{2}} d v

定数を除く:

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

= \frac{1}{6} (- \int \frac{v ^{5}}{1 + v ^{2}} d v)

長除法

\frac{v ^{5}}{1 + v ^{2}} : v^{3} - v + \frac{v}{v ^{2} + 1}

= \frac{1}{6} (- \int v^{3} - v + \frac{v}{v ^{2} + 1} 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}{6} (- (\int v^{3} d v - \int v d v + \int \frac{v}{v ^{2} + 1} d v))

\int v^{3} d v = \frac{v ^{4}}{4}

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

\int \frac{v}{v ^{2} + 1} d v = \frac{1}{2} ln v^{2} + 1

= \frac{1}{6} (- (\frac{v ^{4}}{4} - \frac{v ^{2}}{2} + \frac{1}{2} ln v^{2} + 1))

代用を戻す

= \frac{1}{6} (- (\frac{cot ^{4} ( 6 x )}{4} - \frac{cot ^{2} ( 6 x )}{2} + \frac{1}{2} ln cot^{2} (6 x) + 1))

簡素化

\frac{1}{6} (- (\frac{cot ^{4} ( 6 x )}{4} - \frac{cot ^{2} ( 6 x )}{2} + \frac{1}{2} ln cot^{2} (6 x) + 1)) : - \frac{1}{6} (- \frac{1}{2} cot^{2} (6 x) + \frac{1}{4} cot^{4} (6 x) + \frac{1}{2} ln (csc^{2} (6 x)))

= - \frac{1}{6} (- \frac{1}{2} cot^{2} (6 x) + \frac{1}{4} cot^{4} (6 x) + \frac{1}{2} ln (csc^{2} (6 x)))

定数を解答に追加する

= - \frac{1}{6} (- \frac{1}{2} cot^{2} (6 x) + \frac{1}{4} cot^{4} (6 x) + \frac{1}{2} ln (csc^{2} (6 x))) + C

グラフ

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