ja

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

人気のある微分積分 >

integral of x^5*e^{x^2}

解

\int x^{5} \cdot e^{x^{2}} d x

解

\frac{1}{2} (e^{x^{2}} x^{4} - 2 (e^{x^{2}} x^{2} - e^{x^{2}})) + C

解答ステップ

\int x^{5} e^{x^{2}} d x

u置換積分法を適用する

= \int \frac{e ^{u^{\frac{2}{5}}} u ^{\frac{1}{5}}}{5} d u

定数を除く:

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

= \frac{1}{5} \cdot \int e^{u^{\frac{2}{5}}} u^{\frac{1}{5}} d u

u置換積分法を適用する

= \frac{1}{5} \cdot \int \frac{5 e ^{v} v ^{2}}{2} d v

定数を除く:

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

= \frac{1}{5} \cdot \frac{5}{2} \cdot \int e^{v} v^{2} d v

部分積分を適用する

= \frac{1}{5} \cdot \frac{5}{2} (v^{2} e^{v} - \int 2 v e^{v} d v)

\int 2 v e^{v} d v = 2 (e^{v} v - e^{v})

= \frac{1}{5} \cdot \frac{5}{2} (v^{2} e^{v} - 2 (e^{v} v - e^{v}))

代用を戻す

= \frac{1}{5} \cdot \frac{5}{2} (((x^{5})^{\frac{2}{5}})^{2} e^{(x^{5})^{\frac{2}{5}}} - 2 (e^{(x^{5})^{\frac{2}{5}}} (x^{5})^{\frac{2}{5}} - e^{(x^{5})^{\frac{2}{5}}}))

簡素化

\frac{1}{5} \cdot \frac{5}{2} (((x^{5})^{\frac{2}{5}})^{2} e^{(x^{5})^{\frac{2}{5}}} - 2 (e^{(x^{5})^{\frac{2}{5}}} (x^{5})^{\frac{2}{5}} - e^{(x^{5})^{\frac{2}{5}}})) : \frac{1}{2} (e^{x^{2}} x^{4} - 2 (e^{x^{2}} x^{2} - e^{x^{2}}))

= \frac{1}{2} (e^{x^{2}} x^{4} - 2 (e^{x^{2}} x^{2} - e^{x^{2}}))

定数を解答に追加する

= \frac{1}{2} (e^{x^{2}} x^{4} - 2 (e^{x^{2}} x^{2} - e^{x^{2}})) + C

グラフ

人気の例

integral of 2^{x+3}

\int 2^{x + 3} d x

integral of tan^5(3x)sec^3(3x)

\int tan^{5} (3 x) sec^{3} (3 x) d x

integral of (x^2-3x+5)^{3/2}(2x-3)

\int (x^{2} - 3 x + 5)^{\frac{3}{2}} (2 x - 3) d x

integral of x^2sin(x^3)cos(x^3)

\int x^{2} sin (x^{3}) cos (x^{3}) d x

積分記号-cos(x)

in t e g r a l - cos (x)