integral from 0 to x of 6e^{x+2y}

解答

\int_{0}^{x} 6 e^{x + 2 y} d y

3 (e^{3 x} - e^{x})

求解步骤

\int_{0}^{x} 6 e^{x + 2 y} d y

提出常数:

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

= 6 \cdot \int_{0}^{x} e^{x + 2 y} d y

使用换元积分法

= 6 \cdot \int_{x}^{3 x} e^{u} \frac{1}{2} d u

提出常数:

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

= 6 \cdot \frac{1}{2} \cdot \int_{x}^{3 x} e^{u} d u

使用积分常见定则:

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

= 6 \cdot \frac{1}{2} [e^{u}]_{x}^{3 x}

化简

6 \cdot \frac{1}{2} [e^{u}]_{x}^{3 x} : 3 [e^{u}]_{x}^{3 x}

= 3 [e^{u}]_{x}^{3 x}

带入上下限计算后得:

e^{3 x} - e^{x}

= 3 (e^{3 x} - e^{x})

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