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人気のある微分積分 >

integral of 2xe^{x^2+y^2}

解

\int 2 x e^{x^{2} + y^{2}} d x

解

e^{y^{2} + x^{2}} + C

解答ステップ

\int 2 x e^{x^{2} + y^{2}} d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int e^{y^{2} + x^{2}} x d x

u置換積分法を適用する

= 2 \cdot \int \frac{e ^{y^{2} + u}}{2} d u

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \frac{1}{2} \cdot \int e^{y^{2} + u} d u

簡素化

e^{y^{2} + u} : e^{y^{2}} e^{u}

= 2 \cdot \frac{1}{2} \cdot \int e^{y^{2}} e^{u} d u

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \frac{1}{2} e^{y^{2}} \cdot \int e^{u} d u

共通積分を使用する:

\int e^{u} d u = e^{u}

= 2 \cdot \frac{1}{2} e^{y^{2}} e^{u}

代用を戻す

u = x^{2}

= 2 \cdot \frac{1}{2} e^{y^{2}} e^{x^{2}}

簡素化

2 \cdot \frac{1}{2} e^{y^{2}} e^{x^{2}} : e^{y^{2} + x^{2}}

= e^{y^{2} + x^{2}}

定数を解答に追加する

= e^{y^{2} + x^{2}} + C

グラフ

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