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受欢迎的微积分 >

integral of (e^{4x})/(1+e^{8x)}

解答

\int \frac{e ^{4 x}}{1 + e ^{8 x}} d x

解答

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

求解步骤

\int \frac{e ^{4 x}}{1 + e ^{8 x}} d x

使用换元积分法

= \int \frac{1}{8 u u - 1} d u

提出常数:

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

= \frac{1}{8} \cdot \int \frac{1}{u u - 1} d u

使用换元积分法

= \frac{1}{8} \cdot \int \frac{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}{8} \cdot 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}{8} \cdot 2 arctan (v)

代回

= \frac{1}{8} \cdot 2 arctan (1 + e^{8 x} - 1)

化简

\frac{1}{8} \cdot 2 arctan (1 + e^{8 x} - 1) : \frac{1}{4} arctan (e^{4 x})

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

解答补常数

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

作图

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