integral from 1 to 3 of (5^{2x}-2^{5x})

解

\int_{1}^{3} (5^{2 x} - 2^{5 x}) d x

\frac{7800}{ln ( 5 )} - \frac{32736}{5 ln ( 2 )}

十進法表記

- 4599.20048 \dots

解答ステップ

\int_{1}^{3} 5^{2 x} - 2^{5 x} d x

簡素化

5^{2 x} - 2^{5 x} : 2 5^{x} - 3 2^{x}

= \int_{1}^{3} 2 5^{x} - 3 2^{x} d x

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{1}^{3} 2 5^{x} d x - \int_{1}^{3} 3 2^{x} d x

\int_{1}^{3} 2 5^{x} d x = \frac{7800}{ln ( 5 )}

\int_{1}^{3} 3 2^{x} d x = \frac{32736}{5 ln ( 2 )}

= \frac{7800}{ln ( 5 )} - \frac{32736}{5 ln ( 2 )}