integral of ((2-sqrt(x))^7)/(sqrt(x))

解答

\int \frac{( 2 - x ) ^{7}}{x} d x

- \frac{x ^{4}}{4} + 4 x^{\frac{7}{2}} - 28 x^{3} + 112 x^{\frac{5}{2}} - 280 x^{2} + 448 x^{\frac{3}{2}} - 448 x + 256 x + C

求解步骤

\int \frac{( 2 - x ) ^{7}}{x} d x

长除

\frac{( 2 - x ) ^{7}}{x} : - x^{3} + 14 (x)^{5} - 84 x^{2} + 280 (x)^{3} - 560 x + 672 x - 448 + \frac{128}{x}

= \int - x^{3} + 14 (x)^{5} - 84 x^{2} + 280 (x)^{3} - 560 x + 672 x - 448 + \frac{128}{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 14 (x)^{5} d x - \int 84 x^{2} d x + \int 280 (x)^{3} d x - \int 560 x d x + \int 672 x d x - \int 448 d x + \int \frac{128}{x} d x

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

\int 14 (x)^{5} d x = 4 x^{\frac{7}{2}}

\int 84 x^{2} d x = 28 x^{3}

\int 280 (x)^{3} d x = 112 x^{\frac{5}{2}}

\int 560 x d x = 280 x^{2}

\int 672 x d x = 448 x^{\frac{3}{2}}

\int 448 d x = 448 x

\int \frac{128}{x} d x = 256 x

= - \frac{x ^{4}}{4} + 4 x^{\frac{7}{2}} - 28 x^{3} + 112 x^{\frac{5}{2}} - 280 x^{2} + 448 x^{\frac{3}{2}} - 448 x + 256 x

解答补常数

= - \frac{x ^{4}}{4} + 4 x^{\frac{7}{2}} - 28 x^{3} + 112 x^{\frac{5}{2}} - 280 x^{2} + 448 x^{\frac{3}{2}} - 448 x + 256 x + C

\frac{d}{d y} (e^{x} tan (y))

\frac{d x}{d t} = t - x

\frac{d}{d x} (3 cos^{3} (x))

\int (5 x - 2)^{7} d x

\frac{\partial}{\partial y} (\frac{y}{1 + x ^{2} y ^{2}})