integral of 5x(9-x^{-9})

解答

\int 5 x (9 - x^{- 9}) d x

5 (\frac{9 x ^{2}}{2} + \frac{1}{7 x ^{7}}) + C

求解步骤

\int 5 x (9 - x^{- 9}) d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int x (9 - x^{- 9}) d x

乘开

x (9 - x^{- 9}) : 9 x - \frac{1}{x ^{8}}

= 5 \cdot \int 9 x - \frac{1}{x ^{8}} 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

= 5 (\int 9 x d x - \int \frac{1}{x ^{8}} d x)

\int 9 x d x = \frac{9 x ^{2}}{2}

\int \frac{1}{x ^{8}} d x = - \frac{1}{7 x ^{7}}

= 5 (\frac{9 x ^{2}}{2} - (- \frac{1}{7 x ^{7}}))

化简

= 5 (\frac{9 x ^{2}}{2} + \frac{1}{7 x ^{7}})

解答补常数

= 5 (\frac{9 x ^{2}}{2} + \frac{1}{7 x ^{7}}) + C

\int cos (2 x + 4) d x

\int sec (x) (7 sec (x) + 4 tan (x)) d x

\int \frac{x}{7 + x ^{2}} d x

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

\int \frac{1}{sin ^{2} ( 2 x ) cos ^{4} ( 2 x )} d x