integral from 0 to 2 of 3^x(1-2^x)

解

\int_{0}^{2} 3^{x} (1 - 2^{x}) d x

\frac{8}{ln ( 3 )} - \frac{35}{ln ( 6 )}

十進法表記

- 12.25195 \dots

解答ステップ

\int_{0}^{2} 3^{x} (1 - 2^{x}) d x

拡張

3^{x} (1 - 2^{x}) : 3^{x} - 6^{x}

= \int_{0}^{2} 3^{x} - 6^{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_{0}^{2} 3^{x} d x - \int_{0}^{2} 6^{x} d x

\int_{0}^{2} 3^{x} d x = \frac{8}{ln ( 3 )}

\int_{0}^{2} 6^{x} d x = \frac{35}{ln ( 6 )}

= \frac{8}{ln ( 3 )} - \frac{35}{ln ( 6 )}