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受欢迎的微积分 >

integral of (a^xln(a))/(1+a^{2x)}

解答

\int \frac{a ^{x} ln ( a )}{1 + a ^{2 x}} d x

解答

arctan (a^{x}) + C

求解步骤

\int \frac{a ^{x} ln ( a )}{1 + a ^{2 x}} d x

提出常数:

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

= ln (a) \cdot \int \frac{a ^{x}}{1 + a ^{2 x}} d x

使用换元积分法

= ln (a) \cdot \int \frac{1}{2 a ^{\frac{l n ( u - 1 )}{2 l n ( a )}} u ln ( a )} d u

提出常数:

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

= ln (a) \frac{1}{2 ln ( a )} \cdot \int \frac{1}{a ^{\frac{l n ( u - 1 )}{2 l n ( a )}} u} d u

使用换元积分法

= ln (a) \frac{1}{2 ln ( a )} \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

= ln (a) \frac{1}{2 ln ( a )} \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)

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

代回

= ln (a) \frac{1}{2 ln ( a )} \cdot 2 arctan (a^{\frac{l n ( 1 + a ^{2 x} - 1 )}{2 l n ( a )}})

化简

ln (a) \frac{1}{2 ln ( a )} \cdot 2 arctan (a^{\frac{l n ( 1 + a ^{2 x} - 1 )}{2 l n ( a )}}) : arctan (a^{x})

= arctan (a^{x})

解答补常数

= arctan (a^{x}) + C

作图

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