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人気のある微分積分 >

integral of (sin(x))/(cos^2(x)+cos(x)-2)

解

\int \frac{sin ( x )}{cos ^{2} ( x ) + cos ( x ) - 2} d x

解

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

解答ステップ

\int \frac{sin ( x )}{cos ^{2} ( x ) + cos ( x ) - 2} d x

u置換積分法を適用する

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

定数を除く:

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

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

平方完成する

u^{2} + u - 2 : (u + \frac{1}{2})^{2} - \frac{9}{4}

= - \int \frac{1}{( u + \frac{1}{2} ) ^{2} - \frac{9}{4}} d u

u置換積分法を適用する

= - \int \frac{4}{4 v ^{2} - 9} d v

定数を除く:

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

= - 4 \cdot \int \frac{1}{4 v ^{2} - 9} d v

置換積分法を適用する

= - 4 \cdot \int \frac{1}{6 ( w ^{2} - 1 )} d w

定数を除く:

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

= - 4 \cdot \frac{1}{6} \cdot \int \frac{1}{w ^{2} - 1} d w

因数

w^{2} - 1 : - (- w^{2} + 1)

= - 4 \cdot \frac{1}{6} \cdot \int \frac{1}{- ( - w ^{2} + 1 )} d w

定数を除く:

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

= - 4 \cdot \frac{1}{6} (- \int \frac{1}{- w ^{2} + 1} d w)

共通積分を使用する:

\int \frac{1}{- w ^{2} + 1} d w = \frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}

= - 4 \cdot \frac{1}{6} (- (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}))

代用を戻す

= - 4 \cdot \frac{1}{6} (- (\frac{ln \frac{2}{3} ( cos ( x ) + \frac{1}{2} ) + 1}{2} - \frac{ln \frac{2}{3} ( cos ( x ) + \frac{1}{2} ) - 1}{2}))

簡素化

- 4 \cdot \frac{1}{6} (- (\frac{ln \frac{2}{3} ( cos ( x ) + \frac{1}{2} ) + 1}{2} - \frac{ln \frac{2}{3} ( cos ( x ) + \frac{1}{2} ) - 1}{2})) : \frac{1}{3} (ln \frac{4}{3} + \frac{2}{3} cos (x) - ln - \frac{2}{3} + \frac{2}{3} cos (x))

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

定数を解答に追加する

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

グラフ

人気の例

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