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人気のある微分積分 >

integral of 3cot^5(x)

解

\int 3 cot^{5} (x) d x

解

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

解答ステップ

\int 3 cot^{5} (x) d x

定数を除く:

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

= 3 \cdot \int cot^{5} (x) d x

u置換積分法を適用する

= 3 \cdot \int - \frac{u ^{5}}{1 + u ^{2}} d u

定数を除く:

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

= 3 (- \int \frac{u ^{5}}{1 + u ^{2}} d u)

長除法

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

= 3 (- \int u^{3} - u + \frac{u}{u ^{2} + 1} 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

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

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

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

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

= 3 (- (\frac{u ^{4}}{4} - \frac{u ^{2}}{2} + \frac{1}{2} ln u^{2} + 1))

代用を戻す

u = cot (x)

= 3 (- (\frac{cot ^{4} ( x )}{4} - \frac{cot ^{2} ( x )}{2} + \frac{1}{2} ln cot^{2} (x) + 1))

簡素化

3 (- (\frac{cot ^{4} ( x )}{4} - \frac{cot ^{2} ( x )}{2} + \frac{1}{2} ln cot^{2} (x) + 1)) : - 3 (- \frac{1}{2} cot^{2} (x) + \frac{1}{4} cot^{4} (x) + \frac{1}{2} ln (csc^{2} (x)))

= - 3 (- \frac{1}{2} cot^{2} (x) + \frac{1}{4} cot^{4} (x) + \frac{1}{2} ln (csc^{2} (x)))

定数を解答に追加する

= - 3 (- \frac{1}{2} cot^{2} (x) + \frac{1}{4} cot^{4} (x) + \frac{1}{2} ln (csc^{2} (x))) + C

グラフ

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