integral of (8x^3-12x^2+7x+2)/(2x-2)

解答

\int \frac{8 x ^{3} - 12 x ^{2} + 7 x + 2}{2 x - 2} d x

\frac{4 x ^{3}}{3} - x^{2} + \frac{3}{2} x + \frac{5}{2} ln ∣ 2 x - 2 ∣ + C

求解步骤

\int \frac{8 x ^{3} - 12 x ^{2} + 7 x + 2}{2 x - 2} d x

长除

\frac{8 x ^{3} - 12 x ^{2} + 7 x + 2}{2 x - 2} : 4 x^{2} - 2 x + \frac{3}{2} + \frac{5}{2 x - 2}

= \int 4 x^{2} - 2 x + \frac{3}{2} + \frac{5}{2 x - 2} 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 4 x^{2} d x - \int 2 x d x + \int \frac{3}{2} d x + \int \frac{5}{2 x - 2} d x

\int 4 x^{2} d x = \frac{4 x ^{3}}{3}

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

\int \frac{3}{2} d x = \frac{3}{2} x

\int \frac{5}{2 x - 2} d x = \frac{5}{2} ln ∣ 2 x - 2 ∣

= \frac{4 x ^{3}}{3} - x^{2} + \frac{3}{2} x + \frac{5}{2} ln ∣ 2 x - 2 ∣

解答补常数

= \frac{4 x ^{3}}{3} - x^{2} + \frac{3}{2} x + \frac{5}{2} ln ∣ 2 x - 2 ∣ + C

\int \frac{6}{x} + \frac{5}{x ^{2}} d x

\int \frac{u + 2}{5 u + 6} d u

\int \frac{1}{( 2 y + 3 )} d y

\int tan (x) sec^{4} (x) d x

\int \frac{9 t ^{2} - 26 t - 5}{3 t ^{2} - 5 t - 2} d t