integral from 3 to 4 of f(2^x)2^x

解

\int_{3}^{4} f (2^{x}) 2^{x} d x

\frac{96 f}{ln ( 2 )}

解答ステップ

\int_{3}^{4} f \cdot 2^{x} \cdot 2^{x} d x

定数を除く:

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

= f \cdot \int_{3}^{4} 2^{x} \cdot 2^{x} d x

簡素化

2^{x} \cdot 2^{x} : 4^{x}

= f \cdot \int_{3}^{4} 4^{x} d x

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

= f [\frac{4 ^{x}}{ln ( 4 )}]_{3}^{4}

\frac{4 ^{x}}{ln ( 4 )} = \frac{2 ^{2 x - 1}}{ln ( 2 )}

= f [\frac{2 ^{2 x - 1}}{ln ( 2 )}]_{3}^{4}

限度を計算する:

\frac{96}{ln ( 2 )}

= f \frac{96}{ln ( 2 )}

簡素化

= \frac{96 f}{ln ( 2 )}