integral of 5x-5^x

解

\int 5 x - 5^{x} d x

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

解答ステップ

\int 5 x - 5^{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 5 x d x - \int 5^{x} d x

\int 5 x d x = \frac{5 x ^{2}}{2}

\int 5^{x} d x = \frac{5 ^{x}}{ln ( 5 )}

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

定数を解答に追加する

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