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人気のある微分積分 >

integral of (7^x)/(sqrt(1-7^{2x))}

解

\int \frac{7 ^{x}}{1 - 7 ^{2 x}} d x

解

\frac{arcsin ( 7 ^{x} )}{ln ( 7 )} + C

解答ステップ

\int \frac{7 ^{x}}{1 - 7 ^{2 x}} d x

7^{2 x} = 4 9^{x}

= \int \frac{7 ^{x}}{1 - 4 9 ^{x}} d x

u置換積分法を適用する

= \int - \frac{1}{2 ln ( 7 ) \cdot 7 ^{\frac{l n ( - u + 1 )}{2 l n ( 7 )}} 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 ( 7 )} \cdot \int \frac{1}{7 ^{\frac{l n ( - u + 1 )}{2 l n ( 7 )}} u} d u

u置換積分法を適用する

= - \frac{1}{2 ln ( 7 )} \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 ( 7 )} (- 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 = arcsin (v)

= - \frac{1}{2 ln ( 7 )} (- 2 arcsin (v))

代用を戻す

= - \frac{1}{2 ln ( 7 )} (- 2 arcsin (7^{\frac{l n ( - ( 1 - 4 9 ^{x} ) + 1 )}{2 l n ( 7 )}}))

簡素化

- \frac{1}{2 ln ( 7 )} (- 2 arcsin (7^{\frac{l n ( - ( 1 - 4 9 ^{x} ) + 1 )}{2 l n ( 7 )}})) : \frac{arcsin ( 7 ^{x} )}{ln ( 7 )}

= \frac{arcsin ( 7 ^{x} )}{ln ( 7 )}

定数を解答に追加する

= \frac{arcsin ( 7 ^{x} )}{ln ( 7 )} + C

グラフ

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