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人気のある微分積分 >

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

解

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

解

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

解答ステップ

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

u置換積分法を適用する

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

定数を除く:

\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

u置換積分法を適用する

= \frac{1}{2 ln ( 3 )} \cdot \int \frac{2}{v ^{2} + 1} d v

定数を除く:

\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

共通積分を使用する:

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

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

代用を戻す

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

簡素化

\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 )}

定数を解答に追加する

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

グラフ

人気の例

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