integral of (e^{x^{1/2}})/(x^3)

解答

\int \frac{e ^{x^{\frac{1}{2}}}}{x ^{3}} d x

2 - \frac{e ^{x^{\frac{1}{2}}}}{4 x ^{2}} + \frac{- 2 e ^{x^{\frac{1}{2}}} x + x ^{\frac{3}{2}} ( - e ^{x^{\frac{1}{2}}} + x Ei ( x ^{\frac{1}{2}} ) - e ^{x^{\frac{1}{2}}} x ^{\frac{1}{2}} )}{24 x ^{\frac{5}{2}}} + C

求解步骤

\int \frac{e ^{x^{\frac{1}{2}}}}{x ^{3}} d x

使用换元积分法

= \int \frac{2 e ^{u}}{u ^{5}} d u

提出常数:

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

= 2 \cdot \int \frac{e ^{u}}{u ^{5}} d u

使用分布积分法

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

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

= 2 (- \frac{e ^{u}}{4 u ^{4}} - (- \frac{1}{4} (- \frac{e ^{u}}{3 u ^{3}} + \frac{1}{3} (- \frac{e ^{u}}{2 u ^{2}} + \frac{1}{2} (- \frac{e ^{u}}{u} + Ei (u))))))

u = x^{\frac{1}{2}}

代回

= 2 - \frac{e ^{x^{\frac{1}{2}}}}{4 ( x ^{\frac{1}{2}} ) ^{4}} - - \frac{1}{4} - \frac{e ^{x^{\frac{1}{2}}}}{3 ( x ^{\frac{1}{2}} ) ^{3}} + \frac{1}{3} - \frac{e ^{x^{\frac{1}{2}}}}{2 ( x ^{\frac{1}{2}} ) ^{2}} + \frac{1}{2} (- \frac{e ^{x^{\frac{1}{2}}}}{x ^{\frac{1}{2}}} + Ei (x^{\frac{1}{2}}))

化简

2 - \frac{e ^{x^{\frac{1}{2}}}}{4 ( x ^{\frac{1}{2}} ) ^{4}} - - \frac{1}{4} - \frac{e ^{x^{\frac{1}{2}}}}{3 ( x ^{\frac{1}{2}} ) ^{3}} + \frac{1}{3} - \frac{e ^{x^{\frac{1}{2}}}}{2 ( x ^{\frac{1}{2}} ) ^{2}} + \frac{1}{2} (- \frac{e ^{x^{\frac{1}{2}}}}{x ^{\frac{1}{2}}} + Ei (x^{\frac{1}{2}})) : 2 - \frac{e ^{x^{\frac{1}{2}}}}{4 x ^{2}} + \frac{- 2 e ^{x^{\frac{1}{2}}} x + x ^{\frac{3}{2}} ( - e ^{x^{\frac{1}{2}}} + x Ei ( x ^{\frac{1}{2}} ) - e ^{x^{\frac{1}{2}}} x ^{\frac{1}{2}} )}{24 x ^{\frac{5}{2}}}

= 2 - \frac{e ^{x^{\frac{1}{2}}}}{4 x ^{2}} + \frac{- 2 e ^{x^{\frac{1}{2}}} x + x ^{\frac{3}{2}} ( - e ^{x^{\frac{1}{2}}} + x Ei ( x ^{\frac{1}{2}} ) - e ^{x^{\frac{1}{2}}} x ^{\frac{1}{2}} )}{24 x ^{\frac{5}{2}}}

解答补常数

= 2 - \frac{e ^{x^{\frac{1}{2}}}}{4 x ^{2}} + \frac{- 2 e ^{x^{\frac{1}{2}}} x + x ^{\frac{3}{2}} ( - e ^{x^{\frac{1}{2}}} + x Ei ( x ^{\frac{1}{2}} ) - e ^{x^{\frac{1}{2}}} x ^{\frac{1}{2}} )}{24 x ^{\frac{5}{2}}} + C

\int (- 2 T + x) \cdot e^{- i \cdot z \cdot x} d x

l a pl a ce t r an s f or m (t^{2}) \cdot (sin (2 \cdot t))

x \to 25 lim (\frac{\frac{1}{x} - \frac{1}{5}}{x - 25})

\frac{d ^{2} y}{d x ^{2}} - y = 10 sin^{2} (x)

\frac{d}{d x} (sin (x^{l n (x)}))