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人気のある微分積分 >

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

解

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

解

ln ∣ x - 3 ∣ - \frac{6}{x - 3} - \frac{9}{2 ( x - 3 ) ^{2}} + C

解答ステップ

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

u置換積分法を適用する

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

拡張

\frac{( u + 3 ) ^{2}}{u ^{3}} : \frac{1}{u} + \frac{6}{u ^{2}} + \frac{9}{u ^{3}}

= \int \frac{1}{u} + \frac{6}{u ^{2}} + \frac{9}{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

= \int \frac{1}{u} d u + \int \frac{6}{u ^{2}} d u + \int \frac{9}{u ^{3}} d u

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{6}{u ^{2}} d u = - \frac{6}{u}

\int \frac{9}{u ^{3}} d u = - \frac{9}{2 u ^{2}}

= ln ∣ u ∣ - \frac{6}{u} - \frac{9}{2 u ^{2}}

代用を戻す

u = x - 3

= ln ∣ x - 3 ∣ - \frac{6}{x - 3} - \frac{9}{2 ( x - 3 ) ^{2}}

定数を解答に追加する

= ln ∣ x - 3 ∣ - \frac{6}{x - 3} - \frac{9}{2 ( x - 3 ) ^{2}} + C

グラフ

人気の例

(d^2y)/(dx^2)+(dy)/(dx)=0

\frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} = 0

tangent f(x)=x^3-7x^2,\at x=0

t an g e n t f (x) = x^{3} - 7 x^{2}, at x = 0

integral of (x^6)/(sqrt(x^{14)-7)}

\int \frac{x ^{6}}{x ^{14} - 7} d x

integral of 4cos^5(x)sin^4(x)

\int 4 cos^{5} (x) sin^{4} (x) d x

limit as x approaches 8-of 7x-7

x \to 8 - lim (7 x - 7)