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

解答

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

\frac{x ^{2}}{4} + \frac{3}{4} x + \frac{9}{8} ln ∣ 2 x - 3 ∣ + C

求解步骤

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

长除

\frac{x ^{2}}{2 x - 3} : \frac{x}{2} + \frac{3}{4} + \frac{9}{4 ( 2 x - 3 )}

= \int \frac{x}{2} + \frac{3}{4} + \frac{9}{4 ( 2 x - 3 )} d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x}{2} d x + \int \frac{3}{4} d x + \int \frac{9}{4 ( 2 x - 3 )} d x

\int \frac{x}{2} d x = \frac{x ^{2}}{4}

\int \frac{3}{4} d x = \frac{3}{4} x

\int \frac{9}{4 ( 2 x - 3 )} d x = \frac{9}{8} ln ∣ 2 x - 3 ∣

= \frac{x ^{2}}{4} + \frac{3}{4} x + \frac{9}{8} ln ∣ 2 x - 3 ∣

解答补常数

= \frac{x ^{2}}{4} + \frac{3}{4} x + \frac{9}{8} ln ∣ 2 x - 3 ∣ + C

\int 5 \frac{3}{4} d x

\int (2 + 2 s) e^{4 s + 2 s^{2}} d s

\int (e^{- 4 x}) d x

\int (sinh (5 x) + cosh (3 x)) d x

\int tan^{3} (7 x) d x