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人気のある微分積分 >

integral of 1/(5sin(x)+3)

解

\int \frac{1}{5 sin ( x ) + 3} d x

解

- \frac{1}{4} (ln \frac{9}{4} + \frac{3}{4} tan (\frac{x}{2}) - ln \frac{1}{4} + \frac{3}{4} tan (\frac{x}{2})) + C

解答ステップ

\int \frac{1}{5 sin ( x ) + 3} d x

u置換積分法を適用する

= \int \frac{2}{10 u + 3 ( 1 + u ^{2} )} d u

定数を除く:

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

= 2 \cdot \int \frac{1}{10 u + 3 ( 1 + u ^{2} )} d u

拡張

3 (1 + u^{2}) : 3 + 3 u^{2}

= 2 \cdot \int \frac{1}{10 u + 3 + 3 u ^{2}} d u

平方完成する

10 u + 3 + 3 u^{2} : 3 (u + \frac{5}{3})^{2} - \frac{16}{3}

= 2 \cdot \int \frac{1}{3 ( u + \frac{5}{3} ) ^{2} - \frac{16}{3}} d u

u置換積分法を適用する

= 2 \cdot \int \frac{3}{9 v ^{2} - 16} d v

定数を除く:

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

= 2 \cdot 3 \cdot \int \frac{1}{9 v ^{2} - 16} d v

置換積分法を適用する

= 2 \cdot 3 \cdot \int \frac{1}{12 ( w ^{2} - 1 )} d w

定数を除く:

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

= 2 \cdot 3 \cdot \frac{1}{12} \cdot \int \frac{1}{w ^{2} - 1} d w

因数

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

= 2 \cdot 3 \cdot \frac{1}{12} \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

= 2 \cdot 3 \cdot \frac{1}{12} (- \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}

= 2 \cdot 3 \cdot \frac{1}{12} (- (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}))

代用を戻す

= 2 \cdot 3 \cdot \frac{1}{12} (- (\frac{ln \frac{3}{4} ( tan ( \frac{x}{2} ) + \frac{5}{3} ) + 1}{2} - \frac{ln \frac{3}{4} ( tan ( \frac{x}{2} ) + \frac{5}{3} ) - 1}{2}))

簡素化

2 \cdot 3 \cdot \frac{1}{12} (- (\frac{ln \frac{3}{4} ( tan ( \frac{x}{2} ) + \frac{5}{3} ) + 1}{2} - \frac{ln \frac{3}{4} ( tan ( \frac{x}{2} ) + \frac{5}{3} ) - 1}{2})) : - \frac{1}{4} (ln \frac{9}{4} + \frac{3}{4} tan (\frac{x}{2}) - ln \frac{1}{4} + \frac{3}{4} tan (\frac{x}{2}))

= - \frac{1}{4} (ln \frac{9}{4} + \frac{3}{4} tan (\frac{x}{2}) - ln \frac{1}{4} + \frac{3}{4} tan (\frac{x}{2}))

定数を解答に追加する

= - \frac{1}{4} (ln \frac{9}{4} + \frac{3}{4} tan (\frac{x}{2}) - ln \frac{1}{4} + \frac{3}{4} tan (\frac{x}{2})) + C

グラフ

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