integral of (w+rs-1/c e^{(r-p)t})

解

\int (w + rs - \frac{1}{c} e^{(r - p) t}) d t

wt - e^{t (r - p)} \frac{1}{c ( r - p )} + rs t + C

解答ステップ

\int w + rs - \frac{1}{c} e^{(r - p) 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 w d t - \int \frac{1}{c} e^{(r - p) t} d t + \int rs d t

\int w d t = wt

\int \frac{1}{c} e^{(r - p) t} d t = e^{t (r - p)} \frac{1}{c ( r - p )}

\int rs d t = rs t

= wt - e^{t (r - p)} \frac{1}{c ( r - p )} + rs t

定数を解答に追加する

= wt - e^{t (r - p)} \frac{1}{c ( r - p )} + rs t + C