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

解答

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

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

求解步骤

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

使用分布积分法

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

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

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

化简

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

解答补常数

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

\int_{0}^{π} (sin (x)) d x

\int (x 7 - x^{2}) d x

x \to 4 lim (x^{3} - c x)

f (x) = \frac{( x - 2 )}{x}

\frac{d}{d x} (e^{\frac{1}{x - 1}})