integral of x^3(8x+sqrt(x))

解答

\int x^{3} (8 x + x) d x

\frac{8 x ^{5}}{5} + \frac{2}{9} x^{\frac{9}{2}} + C

求解步骤

\int x^{3} (8 x + x) d x

乘开

x^{3} (8 x + x) : 8 x^{4} + x^{\frac{7}{2}}

= \int 8 x^{4} + x^{\frac{7}{2}} 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 8 x^{4} d x + \int x^{\frac{7}{2}} d x

\int 8 x^{4} d x = \frac{8 x ^{5}}{5}

\int x^{\frac{7}{2}} d x = \frac{2}{9} x^{\frac{9}{2}}

= \frac{8 x ^{5}}{5} + \frac{2}{9} x^{\frac{9}{2}}

解答补常数

= \frac{8 x ^{5}}{5} + \frac{2}{9} x^{\frac{9}{2}} + C

\int r 25 - r^{2} d r

\int \frac{9}{1 + 9 u ^{2}} d u

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

\int 6 - \frac{6}{x ^{2}} d x

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