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人気のある微分積分 >

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

解

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

解

- \frac{1}{2 ( x - 3 ) ^{2}} - \frac{2}{( x - 3 ) ^{3}} - \frac{9}{4 ( x - 3 ) ^{4}} + C

解答ステップ

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

u置換積分法を適用する

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

拡張

\frac{( u + 3 ) ^{2}}{u ^{5}} : \frac{1}{u ^{3}} + \frac{6}{u ^{4}} + \frac{9}{u ^{5}}

= \int \frac{1}{u ^{3}} + \frac{6}{u ^{4}} + \frac{9}{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

= \int \frac{1}{u ^{3}} d u + \int \frac{6}{u ^{4}} d u + \int \frac{9}{u ^{5}} d u

\int \frac{1}{u ^{3}} d u = - \frac{1}{2 u ^{2}}

\int \frac{6}{u ^{4}} d u = - \frac{2}{u ^{3}}

\int \frac{9}{u ^{5}} d u = - \frac{9}{4 u ^{4}}

= - \frac{1}{2 u ^{2}} - \frac{2}{u ^{3}} - \frac{9}{4 u ^{4}}

代用を戻す

u = x - 3

= - \frac{1}{2 ( x - 3 ) ^{2}} - \frac{2}{( x - 3 ) ^{3}} - \frac{9}{4 ( x - 3 ) ^{4}}

定数を解答に追加する

= - \frac{1}{2 ( x - 3 ) ^{2}} - \frac{2}{( x - 3 ) ^{3}} - \frac{9}{4 ( x - 3 ) ^{4}} + C

グラフ

人気の例

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

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

integral of x^2(x^3+8)^{10}

\int x^{2} (x^{3} + 8)^{10} d x

integral of 1/((x+3)sqrt(x+1))

\int \frac{1}{( x + 3 ) x + 1} d x

integral of x/((1-x)^2(1+x))

\int \frac{x}{( 1 - x ) ^{2} ( 1 + x )} d x

integral of (e^{3x})/(3-e^{3x)}

\int \frac{e ^{3 x}}{3 - e ^{3 x}} d x