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人気のある微分積分 >

integral of (sec(x))/(sec(x)+tan(x))

解

\int \frac{sec ( x )}{sec ( x ) + tan ( x )} d x

解

- 2 (- \frac{1}{( 1 + tan ( \frac{x}{2} ) ) ^{2}} + \frac{2}{1 + tan ( \frac{x}{2} )} + \frac{tan ( \frac{x}{2} )}{- 2 tan ( \frac{x}{2} ) - sec ^{2} ( \frac{x}{2} )}) + C

解答ステップ

\int \frac{sec ( x )}{sec ( x ) + tan ( x )} d x

u置換積分法を適用する

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

部分積分を適用する

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

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

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

代用を戻す

u = tan (\frac{x}{2})

= - 2 (\frac{tan ( \frac{x}{2} )}{- tan ^{2} ( \frac{x}{2} ) - 2 tan ( \frac{x}{2} ) - 1} - (- \frac{2}{tan ( \frac{x}{2} ) + 1} + \frac{1}{( tan ( \frac{x}{2} ) + 1 ) ^{2}}))

簡素化

- 2 (\frac{tan ( \frac{x}{2} )}{- tan ^{2} ( \frac{x}{2} ) - 2 tan ( \frac{x}{2} ) - 1} - (- \frac{2}{tan ( \frac{x}{2} ) + 1} + \frac{1}{( tan ( \frac{x}{2} ) + 1 ) ^{2}})) : - 2 (- \frac{1}{( 1 + tan ( \frac{x}{2} ) ) ^{2}} + \frac{2}{1 + tan ( \frac{x}{2} )} + \frac{tan ( \frac{x}{2} )}{- 2 tan ( \frac{x}{2} ) - sec ^{2} ( \frac{x}{2} )})

= - 2 (- \frac{1}{( 1 + tan ( \frac{x}{2} ) ) ^{2}} + \frac{2}{1 + tan ( \frac{x}{2} )} + \frac{tan ( \frac{x}{2} )}{- 2 tan ( \frac{x}{2} ) - sec ^{2} ( \frac{x}{2} )})

定数を解答に追加する

= - 2 (- \frac{1}{( 1 + tan ( \frac{x}{2} ) ) ^{2}} + \frac{2}{1 + tan ( \frac{x}{2} )} + \frac{tan ( \frac{x}{2} )}{- 2 tan ( \frac{x}{2} ) - sec ^{2} ( \frac{x}{2} )}) + C

グラフ

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