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人気のある微分積分 >

integral of (x^7)/((x^4+3)^5)

解

\int \frac{x ^{7}}{( x ^{4} + 3 ) ^{5}} d x

解

\frac{- 4 x ^{4} - 3}{48 ( x ^{2} + 2 4 3 x + 3 ) ^{4} ( x ^{2} - 2 4 3 x + 3 ) ^{4}} + C

解答ステップ

\int \frac{x ^{7}}{( x ^{4} + 3 ) ^{5}} d x

u置換積分法を適用する

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

定数を除く:

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

= \frac{1}{4} \cdot \int \frac{u - 3}{u ^{5}} d u

拡張

\frac{u - 3}{u ^{5}} : \frac{1}{u ^{4}} - \frac{3}{u ^{5}}

= \frac{1}{4} \cdot \int \frac{1}{u ^{4}} - \frac{3}{u ^{5}} 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{1}{4} (\int \frac{1}{u ^{4}} d u - \int \frac{3}{u ^{5}} d u)

\int \frac{1}{u ^{4}} d u = - \frac{1}{3 u ^{3}}

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

= \frac{1}{4} (- \frac{1}{3 u ^{3}} - (- \frac{3}{4 u ^{4}}))

代用を戻す

u = x^{4} + 3

= \frac{1}{4} (- \frac{1}{3 ( x ^{4} + 3 ) ^{3}} - (- \frac{3}{4 ( x ^{4} + 3 ) ^{4}}))

簡素化

\frac{1}{4} (- \frac{1}{3 ( x ^{4} + 3 ) ^{3}} - (- \frac{3}{4 ( x ^{4} + 3 ) ^{4}})) : \frac{- 4 x ^{4} - 3}{48 ( x ^{2} + 2 4 3 x + 3 ) ^{4} ( x ^{2} - 2 4 3 x + 3 ) ^{4}}

= \frac{- 4 x ^{4} - 3}{48 ( x ^{2} + 2 4 3 x + 3 ) ^{4} ( x ^{2} - 2 4 3 x + 3 ) ^{4}}

定数を解答に追加する

= \frac{- 4 x ^{4} - 3}{48 ( x ^{2} + 2 4 3 x + 3 ) ^{4} ( x ^{2} - 2 4 3 x + 3 ) ^{4}} + C

グラフ

人気の例

integral of (8x^3+2/(x^2))

\int (8 x^{3} + \frac{2}{x ^{2}}) d x

integral of (3/4 x-2/5)^2

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

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\int (7 x^{3} + 4 3 x^{2}) d x

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\int (x^{2} + 1) \cdot e^{x} d x