integral of 2^t(1+5^t)

解

\int 2^{t} (1 + 5^{t}) d t

\frac{2 ^{t}}{ln ( 2 )} + \frac{1 0 ^{t}}{ln ( 10 )} + C

解答ステップ

\int 2^{t} (1 + 5^{t}) d t

拡張

2^{t} (1 + 5^{t}) : 2^{t} + 1 0^{t}

= \int 2^{t} + 1 0^{t} d t

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 2^{t} d t + \int 1 0^{t} d t

\int 2^{t} d t = \frac{2 ^{t}}{ln ( 2 )}

\int 1 0^{t} d t = \frac{1 0 ^{t}}{ln ( 10 )}

= \frac{2 ^{t}}{ln ( 2 )} + \frac{1 0 ^{t}}{ln ( 10 )}

定数を解答に追加する

= \frac{2 ^{t}}{ln ( 2 )} + \frac{1 0 ^{t}}{ln ( 10 )} + C