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人気のある微分積分 >

integral of x^9sqrt(x^5+3)

解

\int x^{9} x^{5} + 3 d x

解

\frac{2}{5} (\frac{1}{5} (x^{5} + 3)^{\frac{5}{2}} - (x^{5} + 3)^{\frac{3}{2}}) + C

解答ステップ

\int x^{9} x^{5} + 3 d x

u置換積分法を適用する

= \int \frac{2 ( u ^{2} - 3 ) u ^{2}}{5} d u

定数を除く:

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

= \frac{2}{5} \cdot \int (u^{2} - 3) u^{2} d u

拡張

(u^{2} - 3) u^{2} : u^{4} - 3 u^{2}

= \frac{2}{5} \cdot \int u^{4} - 3 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

= \frac{2}{5} (\int u^{4} d u - \int 3 u^{2} d u)

\int u^{4} d u = \frac{u ^{5}}{5}

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

= \frac{2}{5} (\frac{u ^{5}}{5} - u^{3})

代用を戻す

u = x^{5} + 3

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

簡素化

\frac{2}{5} (\frac{( x ^{5} + 3 ) ^{5}}{5} - (x^{5} + 3)^{3}) : \frac{2}{5} (\frac{1}{5} (x^{5} + 3)^{\frac{5}{2}} - (x^{5} + 3)^{\frac{3}{2}})

= \frac{2}{5} (\frac{1}{5} (x^{5} + 3)^{\frac{5}{2}} - (x^{5} + 3)^{\frac{3}{2}})

定数を解答に追加する

= \frac{2}{5} (\frac{1}{5} (x^{5} + 3)^{\frac{5}{2}} - (x^{5} + 3)^{\frac{3}{2}}) + C

グラフ

人気の例

derivative of ln((5x^2))

\frac{d}{d x} (ln ((5 x)^{2}))

y^{^{''}} + 7 y^{^{'}} = 0

derivative of cot(3-2x)

\frac{d}{d x} (cot (3 - 2 x))

(d^2y)/(dt^2)+5(dy)/(dt)+6y=1

\frac{d ^{2} y}{d t ^{2}} + 5 \frac{d y}{d t} + 6 y = 1

xy^2y^'=y^3-4x^3,y(1)=4

x y^{2} y^{^{'}} = y^{3} - 4 x^{3}, y (1) = 4