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受欢迎的微积分 >

integral of 1/2 e^{1/2 x}

解答

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

解答

e^{\frac{1}{2} x} + C

求解步骤

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

提出常数:

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

= \frac{1}{2} \cdot \int e^{\frac{1}{2} x} d x

使用换元积分法

= \frac{1}{2} \cdot \int e^{u} \cdot 2 d u

提出常数:

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

= \frac{1}{2} \cdot 2 \cdot \int e^{u} d u

使用积分常见定则:

\int e^{u} d u = e^{u}

= \frac{1}{2} \cdot 2 e^{u}

u = \frac{1}{2} x

代回

= \frac{1}{2} \cdot 2 e^{\frac{1}{2} x}

化简

\frac{1}{2} \cdot 2 e^{\frac{1}{2} x} : e^{\frac{1}{2} x}

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

解答补常数

= e^{\frac{1}{2} x} + C

作图

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