积分 ((1-x^8))/((1-x))

解答

积分

\frac{( 1 - x ^{8} )}{( 1 - x )}

\frac{x ^{8}}{8} + \frac{x ^{7}}{7} + \frac{x ^{6}}{6} + \frac{x ^{5}}{5} + \frac{x ^{4}}{4} + \frac{x ^{3}}{3} + \frac{x ^{2}}{2} + x + C

求解步骤

\int \frac{1 - x ^{8}}{1 - x} d x

长除

\frac{1 - x ^{8}}{1 - x} : x^{7} + x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1

= \int x^{7} + x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1 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 x^{6} d x + \int x^{5} d x + \int x^{4} d x + \int x^{3} d x + \int x^{2} d x + \int x d x + \int 1 d x

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

\int x^{6} d x = \frac{x ^{7}}{7}

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

\int x^{4} d x = \frac{x ^{5}}{5}

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

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

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

\int 1 d x = x

= \frac{x ^{8}}{8} + \frac{x ^{7}}{7} + \frac{x ^{6}}{6} + \frac{x ^{5}}{5} + \frac{x ^{4}}{4} + \frac{x ^{3}}{3} + \frac{x ^{2}}{2} + x

解答补常数

= \frac{x ^{8}}{8} + \frac{x ^{7}}{7} + \frac{x ^{6}}{6} + \frac{x ^{5}}{5} + \frac{x ^{4}}{4} + \frac{x ^{3}}{3} + \frac{x ^{2}}{2} + x + C

in t e g r a l f (8 - 2 x)

in t e g r a l \frac{e ^{x}}{( e ^{x} + 2 )}

in t e g r a l 2 x (4 - x^{2})

in t e g r a l \frac{e ^{1}}{e ^{x} + 1}

in t e g r a l \frac{ln ( 1 + x ^{2} )}{x ^{2}}