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人気のある微分積分 >

integral of x(x+1)^{100}

解

\int x (x + 1)^{100} d x

解

\frac{( x + 1 ) ^{102}}{102} - \frac{( x + 1 ) ^{101}}{101} + C

解答ステップ

\int x (x + 1)^{100} d x

u置換積分法を適用する

= \int (u - 1) u^{100} d u

拡張

(u - 1) u^{100} : u^{101} - u^{100}

= \int u^{101} - u^{100} 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^{101} d u - \int u^{100} d u

\int u^{101} d u = \frac{u ^{102}}{102}

\int u^{100} d u = \frac{u ^{101}}{101}

= \frac{u ^{102}}{102} - \frac{u ^{101}}{101}

代用を戻す

u = x + 1

= \frac{( x + 1 ) ^{102}}{102} - \frac{( x + 1 ) ^{101}}{101}

定数を解答に追加する

= \frac{( x + 1 ) ^{102}}{102} - \frac{( x + 1 ) ^{101}}{101} + C

グラフ

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