integral from 0 to xb of ((1-0.25x)^3)/((1-1/2 x)(1-x)^2)

解

\int_{0}^{x b} \frac{( 1 - 0.25 x ) ^{3}}{( 1 - \frac{1}{2} x ) ( 1 - x ) ^{2}} d x

0.03125 b x - \frac{0.84375}{b x - 1} - 0.25 ln ∣ b x - 2 ∣ - 0.67046 \dots

解答ステップ

\int_{0}^{x b} \frac{( 1 - 0.25 x ) ^{3}}{( 1 - \frac{1}{2} x ) ( 1 - x ) ^{2}} d x

以下の部分分数を得る:

\frac{( 1 - 0.25 x ) ^{3}}{( 1 - \frac{1}{2} x ) ( 1 - x ) ^{2}} : 0.03125 - \frac{0.25}{x - 2} + \frac{0.84375}{( x - 1 ) ^{2}}

= \int_{0}^{x b} 0.03125 - \frac{0.25}{x - 2} + \frac{0.84375}{( x - 1 ) ^{2}} d x

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_{0}^{x b} 0.03125 d x - \int_{0}^{x b} \frac{0.25}{x - 2} d x + \int_{0}^{x b} \frac{0.84375}{( x - 1 ) ^{2}} d x

\int_{0}^{x b} 0.03125 d x = 0.03125 b x

\int_{0}^{x b} \frac{0.25}{x - 2} d x = 0.25 (ln ∣ b x - 2 ∣ - 0.69314 \dots)

\int_{0}^{x b} \frac{0.84375}{( x - 1 ) ^{2}} d x = 0.84375 (- \frac{1}{x b - 1} - 1)

= 0.03125 b x - 0.25 (ln ∣ b x - 2 ∣ - 0.69314 \dots) + 0.84375 (- \frac{1}{x b - 1} - 1)

簡素化

0.03125 b x - 0.25 (ln ∣ b x - 2 ∣ - 0.69314 \dots) + 0.84375 (- \frac{1}{x b - 1} - 1) : 0.03125 b x - \frac{0.84375}{b x - 1} - 0.25 ln ∣ b x - 2 ∣ - 0.67046 \dots

= 0.03125 b x - \frac{0.84375}{b x - 1} - 0.25 ln ∣ b x - 2 ∣ - 0.67046 \dots