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人気のある微分積分 >

integral of x*e^{sqrt(x)}

解

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

解

2 (e^{x} x^{\frac{3}{2}} - 3 (e^{x} x - 2 (e^{x} x - e^{x}))) + C

解答ステップ

\int x e^{x} d x

u置換積分法を適用する

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

定数を除く:

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

= 2 \cdot \int e^{u} u^{3} d u

部分積分を適用する

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

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

= 2 (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))

代用を戻す

u = x

= 2 ((x)^{3} e^{x} - 3 ((x)^{2} e^{x} - 2 (e^{x} x - e^{x})))

簡素化

2 ((x)^{3} e^{x} - 3 ((x)^{2} e^{x} - 2 (e^{x} x - e^{x}))) : 2 (e^{x} x^{\frac{3}{2}} - 3 (e^{x} x - 2 (e^{x} x - e^{x})))

= 2 (e^{x} x^{\frac{3}{2}} - 3 (e^{x} x - 2 (e^{x} x - e^{x})))

定数を解答に追加する

= 2 (e^{x} x^{\frac{3}{2}} - 3 (e^{x} x - 2 (e^{x} x - e^{x}))) + C

グラフ

人気の例

integral of (ycos(x))

\int (y cos (x)) d x

integral of (2sin(x))/(sqrt(cos(x)))

\int \frac{2 sin ( x )}{cos ( x )} d x

integral of (1+cos^2(5)x^2csc^2(5)x^2)6x

\int (1 + cos^{2} (5) x^{2} csc^{2} (5) x^{2}) 6 x d x

integral of xcot^2(2x^2)

\int x cot^{2} (2 x^{2}) d x

integral of xsqrt(3x^2+2)

\int x 3 x^{2} + 2 d x