integral of (x^2+1)/((2x-3)^2)

解答

\int \frac{x ^{2} + 1}{( 2 x - 3 ) ^{2}} d x

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

求解步骤

\int \frac{x ^{2} + 1}{( 2 x - 3 ) ^{2}} d x

使用分布积分法

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

\int - \frac{x}{2 x - 3} d x = - \frac{1}{4} (2 x - 3 + 3 ln ∣ 2 x - 3 ∣)

= - \frac{x ^{2} + 1}{2 ( 2 x - 3 )} - (- \frac{1}{4} (2 x - 3 + 3 ln ∣ 2 x - 3 ∣))

化简

= - \frac{x ^{2} + 1}{2 ( 2 x - 3 )} + \frac{1}{4} (2 x - 3 + 3 ln ∣ 2 x - 3 ∣)

解答补常数

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

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