integral of ((x(x-8))/(x+1))^2

解答

\int (\frac{x ( x - 8 )}{x + 1})^{2} d x

- 8 x^{2} + 172 x + \frac{4}{3} (x + 1)^{3} - \frac{x ^{4}}{x + 1} - \frac{16}{x + 1} - \frac{64 x ^{2}}{x + 1} - 180 ln ∣ x + 1 ∣ - 6 (x + 1)^{2} + 180 + C

求解步骤

\int (\frac{x ( x - 8 )}{x + 1})^{2} d x

乘开

(\frac{x ( x - 8 )}{x + 1})^{2} : \frac{x ^{4} - 16 x ^{3} + 64 x ^{2}}{x ^{2} + 2 x + 1}

= \int \frac{x ^{4} - 16 x ^{3} + 64 x ^{2}}{x ^{2} + 2 x + 1} d x

乘开

\frac{x ^{4} - 16 x ^{3} + 64 x ^{2}}{x ^{2} + 2 x + 1} : \frac{x ^{4}}{x ^{2} + 2 x + 1} - \frac{16 x ^{3}}{x ^{2} + 2 x + 1} + \frac{64 x ^{2}}{x ^{2} + 2 x + 1}

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 ^{4}}{x ^{2} + 2 x + 1} d x - \int \frac{16 x ^{3}}{x ^{2} + 2 x + 1} d x + \int \frac{64 x ^{2}}{x ^{2} + 2 x + 1} d x

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

\int \frac{16 x ^{3}}{x ^{2} + 2 x + 1} d x = 16 (\frac{1}{2} (x + 1)^{2} - 3 (x + 1) + 3 ln ∣ x + 1 ∣ + \frac{1}{x + 1})

\int \frac{64 x ^{2}}{x ^{2} + 2 x + 1} d x = 64 (- \frac{x ^{2}}{x + 1} + 2 (x + 1 - ln ∣ x + 1 ∣))

= - \frac{x ^{4}}{x + 1} + 4 (\frac{1}{3} (x + 1)^{3} - \frac{3}{2} (x + 1)^{2} + 3 (x + 1) - ln ∣ x + 1 ∣) - 16 (\frac{1}{2} (x + 1)^{2} - 3 (x + 1) + 3 ln ∣ x + 1 ∣ + \frac{1}{x + 1}) + 64 (- \frac{x ^{2}}{x + 1} + 2 (x + 1 - ln ∣ x + 1 ∣))

化简

- \frac{x ^{4}}{x + 1} + 4 (\frac{1}{3} (x + 1)^{3} - \frac{3}{2} (x + 1)^{2} + 3 (x + 1) - ln ∣ x + 1 ∣) - 16 (\frac{1}{2} (x + 1)^{2} - 3 (x + 1) + 3 ln ∣ x + 1 ∣ + \frac{1}{x + 1}) + 64 (- \frac{x ^{2}}{x + 1} + 2 (x + 1 - ln ∣ x + 1 ∣)) : - 8 x^{2} + 172 x + \frac{4}{3} (x + 1)^{3} - \frac{x ^{4}}{x + 1} - \frac{16}{x + 1} - \frac{64 x ^{2}}{x + 1} - 180 ln ∣ x + 1 ∣ - 6 (x + 1)^{2} + 180

= - 8 x^{2} + 172 x + \frac{4}{3} (x + 1)^{3} - \frac{x ^{4}}{x + 1} - \frac{16}{x + 1} - \frac{64 x ^{2}}{x + 1} - 180 ln ∣ x + 1 ∣ - 6 (x + 1)^{2} + 180

解答补常数

= - 8 x^{2} + 172 x + \frac{4}{3} (x + 1)^{3} - \frac{x ^{4}}{x + 1} - \frac{16}{x + 1} - \frac{64 x ^{2}}{x + 1} - 180 ln ∣ x + 1 ∣ - 6 (x + 1)^{2} + 180 + C

d er i v a t i v e f (x) = 5 x (sin (x) + cos (x))

\int sin (π + 2 y) - sin (2 y) d y

\frac{\partial}{\partial y} (x y e^{y} + e^{y} sin (x))

\frac{\partial}{\partial x} (16 - x^{2} - y^{2})

a re a y = e^{- 2 x} + 1, y = 0, x = 0, x = 3