integral of x(x-5)^2

解答

\int x (x - 5)^{2} d x

\frac{x ^{4}}{4} - \frac{10 x ^{3}}{3} + \frac{25 x ^{2}}{2} + C

求解步骤

\int x (x - 5)^{2} d x

乘开

x (x - 5)^{2} : x^{3} - 10 x^{2} + 25 x

= \int x^{3} - 10 x^{2} + 25 x 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 x^{3} d x - \int 10 x^{2} d x + \int 25 x d x

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

\int 10 x^{2} d x = \frac{10 x ^{3}}{3}

\int 25 x d x = \frac{25 x ^{2}}{2}

= \frac{x ^{4}}{4} - \frac{10 x ^{3}}{3} + \frac{25 x ^{2}}{2}

解答补常数

= \frac{x ^{4}}{4} - \frac{10 x ^{3}}{3} + \frac{25 x ^{2}}{2} + C

d er i v a t i v e f (x) = \frac{x ^{3}}{24} + \frac{2}{x}

\int_{1}^{2} \frac{1}{4 - t ^{2}} d t

t an g e n t - 3 x^{2} + 2 x y + 2 y^{3} = 125, (3, 4)

\frac{d}{d x} y = ln (10 x^{2} - 8)

\frac{d}{d x} (I n x)