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人気のある微分積分 >

integral of ((sin(x))^3)/(1+(cos(x))^3)

解

\int \frac{( sin ( x ) ) ^{3}}{1 + ( cos ( x ) ) ^{3}} d x

解

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

解答ステップ

\int \frac{( sin ( x ) ) ^{3}}{1 + ( cos ( x ) ) ^{3}} d x

u置換積分法を適用する

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

平方完成する

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

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

u置換積分法を適用する

= \int \frac{2 ( 2 v - 1 )}{4 v ^{2} + 3} d v

定数を除く:

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

= 2 \cdot \int \frac{2 v - 1}{4 v ^{2} + 3} d v

拡張

\frac{2 v - 1}{4 v ^{2} + 3} : \frac{2 v}{4 v ^{2} + 3} - \frac{1}{4 v ^{2} + 3}

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

= 2 (\int \frac{2 v}{4 v ^{2} + 3} d v - \int \frac{1}{4 v ^{2} + 3} d v)

\int \frac{2 v}{4 v ^{2} + 3} d v = \frac{1}{4} ln 4 v^{2} + 3

\int \frac{1}{4 v ^{2} + 3} d v = \frac{1}{2 3} arctan (\frac{2}{3} v)

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

代用を戻す

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

簡素化

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

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

定数を解答に追加する

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

グラフ

人気の例

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\frac{d}{d x} (\frac{5}{5 - x})

勾配 (3,1),(1,-8)

s l o p e (3, 1), (1, - 8)

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