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

解答

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

\frac{x ^{8}}{8} + \frac{x ^{6}}{2} + \frac{3 x ^{4}}{4} + \frac{x ^{2}}{2} + C

求解步骤

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

乘开

x (x^{2} + 1)^{3} : x^{7} + 3 x^{5} + 3 x^{3} + x

= \int x^{7} + 3 x^{5} + 3 x^{3} + 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^{7} d x + \int 3 x^{5} d x + \int 3 x^{3} d x + \int x d x

\int x^{7} d x = \frac{x ^{8}}{8}

\int 3 x^{5} d x = \frac{x ^{6}}{2}

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

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

= \frac{x ^{8}}{8} + \frac{x ^{6}}{2} + \frac{3 x ^{4}}{4} + \frac{x ^{2}}{2}

解答补常数

= \frac{x ^{8}}{8} + \frac{x ^{6}}{2} + \frac{3 x ^{4}}{4} + \frac{x ^{2}}{2} + C

x \to π lim (- 3 cos (6 x))

d er i v a t i v e f (x) = 10 e^{x}

d er i v a t i v e \frac{2 x}{e ^{x}}

2 y y^{^{'}} + 4 = y^{2} + 4 x, y (0) = 5

t \to 0 lim (sin (3 t))