integral from x to sqrt(x of)e^{x/y}

解答

\int_{x}^{x} e^{\frac{x}{y}} d y

- x (- \frac{e ^{x}}{x} + Ei (x) + e - Ei (1))

求解步骤

\int_{x}^{x} e^{\frac{x}{y}} d y

使用换元积分法

= \int_{1}^{x^{\frac{1}{2}}} - \frac{x e ^{u}}{u ^{2}} d u

提出常数:

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

= - x \cdot \int_{1}^{x^{\frac{1}{2}}} \frac{e ^{u}}{u ^{2}} d u

使用分布积分法

= - x [- \frac{e ^{u}}{u} - \int - \frac{e ^{u}}{u} d u]_{1}^{x^{\frac{1}{2}}}

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

= - x [- \frac{e ^{u}}{u} - (- Ei (u))]_{1}^{x^{\frac{1}{2}}}

化简

- x [- \frac{e ^{u}}{u} - (- Ei (u))]_{1}^{x^{\frac{1}{2}}} : - x [- \frac{e ^{u}}{u} + Ei (u)]_{1}^{x}

= - x [- \frac{e ^{u}}{u} + Ei (u)]_{1}^{x}

带入上下限计算后得:

- \frac{e ^{x}}{x} + Ei (x) + e - Ei (1)

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

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