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人気のある微分積分 >

integral of (2x+1)/((x+2)^4(x-1)^4)

解

\int \frac{2 x + 1}{( x + 2 ) ^{4} ( x - 1 ) ^{4}} d x

解

128 (\frac{1}{7776 ( 2 x + 4 )} + \frac{1}{2592 ( 2 x + 4 ) ^{2}} + \frac{1}{1296 ( 2 x + 4 ) ^{3}} - \frac{1}{7776 ( 2 x - 2 )} + \frac{1}{2592 ( 2 x - 2 ) ^{2}} - \frac{1}{1296 ( 2 x - 2 ) ^{3}}) + C

解答ステップ

\int \frac{2 x + 1}{( x + 2 ) ^{4} ( x - 1 ) ^{4}} d x

u置換積分法を適用する

= \int \frac{128 u}{( u + 3 ) ^{4} ( u - 3 ) ^{4}} d u

定数を除く:

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

= 128 \cdot \int \frac{u}{( u + 3 ) ^{4} ( u - 3 ) ^{4}} d u

以下の部分分数を得る:

\frac{u}{( u + 3 ) ^{4} ( u - 3 ) ^{4}} : - \frac{1}{7776 ( u + 3 ) ^{2}} - \frac{1}{1296 ( u + 3 ) ^{3}} - \frac{1}{432 ( u + 3 ) ^{4}} + \frac{1}{7776 ( u - 3 ) ^{2}} - \frac{1}{1296 ( u - 3 ) ^{3}} + \frac{1}{432 ( u - 3 ) ^{4}}

= 128 \cdot \int - \frac{1}{7776 ( u + 3 ) ^{2}} - \frac{1}{1296 ( u + 3 ) ^{3}} - \frac{1}{432 ( u + 3 ) ^{4}} + \frac{1}{7776 ( u - 3 ) ^{2}} - \frac{1}{1296 ( u - 3 ) ^{3}} + \frac{1}{432 ( u - 3 ) ^{4}} 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

= 128 (- \int \frac{1}{7776 ( u + 3 ) ^{2}} d u - \int \frac{1}{1296 ( u + 3 ) ^{3}} d u - \int \frac{1}{432 ( u + 3 ) ^{4}} d u + \int \frac{1}{7776 ( u - 3 ) ^{2}} d u - \int \frac{1}{1296 ( u - 3 ) ^{3}} d u + \int \frac{1}{432 ( u - 3 ) ^{4}} d u)

\int \frac{1}{7776 ( u + 3 ) ^{2}} d u = - \frac{1}{7776 ( u + 3 )}

\int \frac{1}{1296 ( u + 3 ) ^{3}} d u = - \frac{1}{2592 ( u + 3 ) ^{2}}

\int \frac{1}{432 ( u + 3 ) ^{4}} d u = - \frac{1}{1296 ( u + 3 ) ^{3}}

\int \frac{1}{7776 ( u - 3 ) ^{2}} d u = - \frac{1}{7776 ( u - 3 )}

\int \frac{1}{1296 ( u - 3 ) ^{3}} d u = - \frac{1}{2592 ( u - 3 ) ^{2}}

\int \frac{1}{432 ( u - 3 ) ^{4}} d u = - \frac{1}{1296 ( u - 3 ) ^{3}}

= 128 (- (- \frac{1}{7776 ( u + 3 )}) - (- \frac{1}{2592 ( u + 3 ) ^{2}}) - (- \frac{1}{1296 ( u + 3 ) ^{3}}) - \frac{1}{7776 ( u - 3 )} - (- \frac{1}{2592 ( u - 3 ) ^{2}}) - \frac{1}{1296 ( u - 3 ) ^{3}})

代用を戻す

u = 2 x + 1

= 128 (- (- \frac{1}{7776 ( 2 x + 1 + 3 )}) - (- \frac{1}{2592 ( 2 x + 1 + 3 ) ^{2}}) - (- \frac{1}{1296 ( 2 x + 1 + 3 ) ^{3}}) - \frac{1}{7776 ( 2 x + 1 - 3 )} - (- \frac{1}{2592 ( 2 x + 1 - 3 ) ^{2}}) - \frac{1}{1296 ( 2 x + 1 - 3 ) ^{3}})

簡素化

128 (- (- \frac{1}{7776 ( 2 x + 1 + 3 )}) - (- \frac{1}{2592 ( 2 x + 1 + 3 ) ^{2}}) - (- \frac{1}{1296 ( 2 x + 1 + 3 ) ^{3}}) - \frac{1}{7776 ( 2 x + 1 - 3 )} - (- \frac{1}{2592 ( 2 x + 1 - 3 ) ^{2}}) - \frac{1}{1296 ( 2 x + 1 - 3 ) ^{3}}) : 128 (\frac{1}{7776 ( 2 x + 4 )} + \frac{1}{2592 ( 2 x + 4 ) ^{2}} + \frac{1}{1296 ( 2 x + 4 ) ^{3}} - \frac{1}{7776 ( 2 x - 2 )} + \frac{1}{2592 ( 2 x - 2 ) ^{2}} - \frac{1}{1296 ( 2 x - 2 ) ^{3}})

= 128 (\frac{1}{7776 ( 2 x + 4 )} + \frac{1}{2592 ( 2 x + 4 ) ^{2}} + \frac{1}{1296 ( 2 x + 4 ) ^{3}} - \frac{1}{7776 ( 2 x - 2 )} + \frac{1}{2592 ( 2 x - 2 ) ^{2}} - \frac{1}{1296 ( 2 x - 2 ) ^{3}})

定数を解答に追加する

= 128 (\frac{1}{7776 ( 2 x + 4 )} + \frac{1}{2592 ( 2 x + 4 ) ^{2}} + \frac{1}{1296 ( 2 x + 4 ) ^{3}} - \frac{1}{7776 ( 2 x - 2 )} + \frac{1}{2592 ( 2 x - 2 ) ^{2}} - \frac{1}{1296 ( 2 x - 2 ) ^{3}}) + C

グラフ

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