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受欢迎的微积分 >

integral of (3^x)/(1+9^x)

解答

\int \frac{3 ^{x}}{1 + 9 ^{x}} d x

解答

\frac{arctan ( 3 ^{x} )}{ln ( 3 )} + C

求解步骤

\int \frac{3 ^{x}}{1 + 9 ^{x}} d x

使用换元积分法

= \int \frac{1}{2 ln ( 3 ) \cdot 3 ^{\frac{l n ( u - 1 )}{2 l n ( 3 )}} u} d u

提出常数:

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

= \frac{1}{2 ln ( 3 )} \cdot \int \frac{1}{3 ^{\frac{l n ( u - 1 )}{2 l n ( 3 )}} u} d u

使用换元积分法

= \frac{1}{2 ln ( 3 )} \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}{2 ln ( 3 )} \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}{2 ln ( 3 )} \cdot 2 arctan (v)

代回

= \frac{1}{2 ln ( 3 )} \cdot 2 arctan (3^{\frac{l n ( 1 + 9 ^{x} - 1 )}{2 l n ( 3 )}})

化简

\frac{1}{2 ln ( 3 )} \cdot 2 arctan (3^{\frac{l n ( 1 + 9 ^{x} - 1 )}{2 l n ( 3 )}}) : \frac{arctan ( 3 ^{x} )}{ln ( 3 )}

= \frac{arctan ( 3 ^{x} )}{ln ( 3 )}

解答补常数

= \frac{arctan ( 3 ^{x} )}{ln ( 3 )} + C

作图

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