integral of ((e^{x/2}-e^{-x/2})^2)/4

解

\int \frac{( e ^{\frac{x}{2}} - e ^{- \frac{x}{2}} ) ^{2}}{4} d x

\frac{1}{4} (e^{x} x - 2 x + e^{- x} x - 2 x cosh (x) + 2 sinh (x)) + C

解答ステップ

\int \frac{( e ^{\frac{x}{2}} - e ^{- \frac{x}{2}} ) ^{2}}{4} d x

定数を除く:

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

= \frac{1}{4} \cdot \int (e^{\frac{x}{2}} - e^{- \frac{x}{2}})^{2} d x

部分積分を適用する

= \frac{1}{4} ((e^{\frac{x}{2}} - e^{- \frac{x}{2}})^{2} x - \int (e^{x} - e^{- x}) x d x)

\int (e^{x} - e^{- x}) x d x = 2 x cosh (x) - 2 sinh (x)

= \frac{1}{4} ((e^{\frac{x}{2}} - e^{- \frac{x}{2}})^{2} x - (2 x cosh (x) - 2 sinh (x)))

拡張

(e^{\frac{x}{2}} - e^{- \frac{x}{2}})^{2} x - (2 x cosh (x) - 2 sinh (x)) : e^{x} x - 2 x + e^{- x} x - 2 x cosh (x) + 2 sinh (x)

= \frac{1}{4} (e^{x} x - 2 x + e^{- x} x - 2 x cosh (x) + 2 sinh (x))

定数を解答に追加する

= \frac{1}{4} (e^{x} x - 2 x + e^{- x} x - 2 x cosh (x) + 2 sinh (x)) + C