integral of e^t(sin(t)+cos(t))

解

\int e^{t} (sin (t) + cos (t)) d t

e^{t} sin (t) + C

解答ステップ

\int e^{t} (sin (t) + cos (t)) d t

拡張

e^{t} (sin (t) + cos (t)) : e^{t} sin (t) + e^{t} cos (t)

= \int e^{t} sin (t) + e^{t} cos (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 e^{t} sin (t) d t + \int e^{t} cos (t) d t

\int e^{t} sin (t) d t = - \frac{e ^{t} cos ( t )}{2} + \frac{e ^{t} sin ( t )}{2}

\int e^{t} cos (t) d t = \frac{e ^{t} sin ( t )}{2} + \frac{e ^{t} cos ( t )}{2}

= (- \frac{e ^{t} cos ( t )}{2} + \frac{e ^{t} sin ( t )}{2}) + \frac{e ^{t} sin ( t )}{2} + \frac{e ^{t} cos ( t )}{2}

簡素化

(- \frac{e ^{t} cos ( t )}{2} + \frac{e ^{t} sin ( t )}{2}) + \frac{e ^{t} sin ( t )}{2} + \frac{e ^{t} cos ( t )}{2} : e^{t} sin (t)

= e^{t} sin (t)

定数を解答に追加する

= e^{t} sin (t) + C