integral of (7x^3-7x+4)/(x^3-x^2)

解答

\int \frac{7 x ^{3} - 7 x + 4}{x ^{3} - x ^{2}} d x

\frac{7 x ^{3} - 7 x + 4}{x ( x - 1 )} + 4 ln ∣ x - 1 ∣ - \frac{4}{x - 1} + 3 ln ∣ x ∣ + \frac{8}{x} + C

求解步骤

\int \frac{7 x ^{3} - 7 x + 4}{x ^{3} - x ^{2}} d x

使用分布积分法

= \frac{7 x ^{3} - 7 x + 4}{x ( x - 1 )} - \int - \frac{7 x ^{3} - 14 x ^{2} + 19 x - 8}{x ^{2} ( x - 1 ) ^{2}} d x

\int - \frac{7 x ^{3} - 14 x ^{2} + 19 x - 8}{x ^{2} ( x - 1 ) ^{2}} d x = - 4 ln ∣ x - 1 ∣ + \frac{4}{x - 1} - 3 ln ∣ x ∣ - \frac{8}{x}

= \frac{7 x ^{3} - 7 x + 4}{x ( x - 1 )} - (- 4 ln ∣ x - 1 ∣ + \frac{4}{x - 1} - 3 ln ∣ x ∣ - \frac{8}{x})

化简

= \frac{7 x ^{3} - 7 x + 4}{x ( x - 1 )} + 4 ln ∣ x - 1 ∣ - \frac{4}{x - 1} + 3 ln ∣ x ∣ + \frac{8}{x}

解答补常数

= \frac{7 x ^{3} - 7 x + 4}{x ( x - 1 )} + 4 ln ∣ x - 1 ∣ - \frac{4}{x - 1} + 3 ln ∣ x ∣ + \frac{8}{x} + C

\int 16 sin^{3} (x) cos^{2} (x) d x

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