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受欢迎的微积分 >

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

解答

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

解答

\frac{1}{4} arctan (\frac{1}{2} e^{2 x}) + C

求解步骤

\int \frac{e ^{2 x}}{e ^{4 x} + 4} d x

使用换元积分法

= \int \frac{1}{2 ( u ^{2} + 4 )} d u

提出常数:

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

= \frac{1}{2} \cdot \int \frac{1}{u ^{2} + 4} d u

使用换元积分法

= \frac{1}{2} \cdot \int \frac{1}{2 ( v ^{2} + 1 )} d v

提出常数:

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

= \frac{1}{2} \cdot \frac{1}{2} \cdot \int \frac{1}{v ^{2} + 1} d v

使用积分常见定则:

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

= \frac{1}{2} \cdot \frac{1}{2} arctan (v)

代回

= \frac{1}{2} \cdot \frac{1}{2} arctan (\frac{e ^{2 x}}{2})

化简

\frac{1}{2} \cdot \frac{1}{2} arctan (\frac{e ^{2 x}}{2}) : \frac{1}{4} arctan (\frac{1}{2} e^{2 x})

= \frac{1}{4} arctan (\frac{1}{2} e^{2 x})

解答补常数

= \frac{1}{4} arctan (\frac{1}{2} e^{2 x}) + C

作图

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