integral of 10^{4x-2}

解

\int 1 0^{4 x - 2} d x

\frac{2 ^{4 x - 4} \cdot 5 ^{4 x - 2}}{ln ( 10 )} + C

解答ステップ

\int 1 0^{4 x - 2} d x

1 0^{4 x - 2} = 1 0^{- 2} \cdot 1 0^{4 x}

= \int 1 0^{- 2} \cdot 1 0^{4 x} d x

定数を除く:

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

= 1 0^{- 2} \cdot \int 1 0^{4 x} d x

1 0^{- 2} = \frac{1}{100}

= \frac{1}{100} \cdot \int 1 0^{4 x} d x

簡素化

1 0^{4 x} : 1000 0^{x}

= \frac{1}{100} \cdot \int 1000 0^{x} d x

\int a^{x} d x = \frac{a ^{x}}{ln a}

= \frac{1}{100} \cdot \frac{1000 0 ^{x}}{ln ( 10000 )}

簡素化

\frac{1}{100} \cdot \frac{1000 0 ^{x}}{ln ( 10000 )} : \frac{2 ^{4 x - 4} \cdot 5 ^{4 x - 2}}{ln ( 10 )}

= \frac{2 ^{4 x - 4} \cdot 5 ^{4 x - 2}}{ln ( 10 )}

定数を解答に追加する

= \frac{2 ^{4 x - 4} \cdot 5 ^{4 x - 2}}{ln ( 10 )} + C