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人気のある微分積分 >

integral of (x^3)/((5x+2)^3)

解

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

解

\frac{125 x ^{3} + 150 x ^{2} - 150 x ^{2} ln ∣ 5 x + 2 ∣ - 120 x ln ∣ 5 x + 2 ∣ - 24 ln ∣ 5 x + 2 ∣ - 12}{625 ( 5 x + 2 ) ^{2}} + C

解答ステップ

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

u置換積分法を適用する

= \int \frac{( u - 2 ) ^{3}}{625 u ^{3}} d u

定数を除く:

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

= \frac{1}{625} \cdot \int \frac{( u - 2 ) ^{3}}{u ^{3}} d u

拡張

\frac{( u - 2 ) ^{3}}{u ^{3}} : 1 - \frac{6}{u} + \frac{12}{u ^{2}} - \frac{8}{u ^{3}}

= \frac{1}{625} \cdot \int 1 - \frac{6}{u} + \frac{12}{u ^{2}} - \frac{8}{u ^{3}} 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}{625} (\int 1 d u - \int \frac{6}{u} d u + \int \frac{12}{u ^{2}} d u - \int \frac{8}{u ^{3}} d u)

\int 1 d u = u

\int \frac{6}{u} d u = 6 ln ∣ u ∣

\int \frac{12}{u ^{2}} d u = - \frac{12}{u}

\int \frac{8}{u ^{3}} d u = - \frac{4}{u ^{2}}

= \frac{1}{625} (u - 6 ln ∣ u ∣ - \frac{12}{u} - (- \frac{4}{u ^{2}}))

代用を戻す

u = 5 x + 2

= \frac{1}{625} (5 x + 2 - 6 ln ∣ 5 x + 2 ∣ - \frac{12}{5 x + 2} - (- \frac{4}{( 5 x + 2 ) ^{2}}))

簡素化

\frac{1}{625} (5 x + 2 - 6 ln ∣ 5 x + 2 ∣ - \frac{12}{5 x + 2} - (- \frac{4}{( 5 x + 2 ) ^{2}})) : \frac{125 x ^{3} + 150 x ^{2} - 150 x ^{2} ln ∣ 5 x + 2 ∣ - 120 x ln ∣ 5 x + 2 ∣ - 24 ln ∣ 5 x + 2 ∣ - 12}{625 ( 5 x + 2 ) ^{2}}

= \frac{125 x ^{3} + 150 x ^{2} - 150 x ^{2} ln ∣ 5 x + 2 ∣ - 120 x ln ∣ 5 x + 2 ∣ - 24 ln ∣ 5 x + 2 ∣ - 12}{625 ( 5 x + 2 ) ^{2}}

定数を解答に追加する

= \frac{125 x ^{3} + 150 x ^{2} - 150 x ^{2} ln ∣ 5 x + 2 ∣ - 120 x ln ∣ 5 x + 2 ∣ - 24 ln ∣ 5 x + 2 ∣ - 12}{625 ( 5 x + 2 ) ^{2}} + C

グラフ

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