integral of (u^2+3-ue^{7-u^2})

解

\int (u^{2} + 3 - u e^{7 - u^{2}}) d u

\frac{u ^{3}}{3} + 3 u + \frac{1}{2} e^{7 - u^{2}} + C

解答ステップ

\int u^{2} + 3 - u e^{7 - u^{2}} d u

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 u^{2} d u + \int 3 d u - \int u e^{7 - u^{2}} d u

\int u^{2} d u = \frac{u ^{3}}{3}

\int 3 d u = 3 u

\int u e^{7 - u^{2}} d u = - \frac{1}{2} e^{7 - u^{2}}

= \frac{u ^{3}}{3} + 3 u - (- \frac{1}{2} e^{7 - u^{2}})

簡素化

= \frac{u ^{3}}{3} + 3 u + \frac{1}{2} e^{7 - u^{2}}

定数を解答に追加する

= \frac{u ^{3}}{3} + 3 u + \frac{1}{2} e^{7 - u^{2}} + C