integral of ((x^2+1)^2)/(sqrt(x))

解答

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

\frac{2}{9} x^{4} x + \frac{4}{5} x^{2} x + 2 x + C

求解步骤

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

使用分布积分法

= x (x^{2} + 1)^{2} - \int \frac{7 x ^{4} + 6 x ^{2} - 1}{2 x} d x

\int \frac{7 x ^{4} + 6 x ^{2} - 1}{2 x} d x = \frac{1}{45} x (35 x^{4} + 54 x^{2} - 45)

= x (x^{2} + 1)^{2} - \frac{1}{45} x (35 x^{4} + 54 x^{2} - 45)

化简

x (x^{2} + 1)^{2} - \frac{1}{45} x (35 x^{4} + 54 x^{2} - 45) : \frac{2}{9} x^{4} x + \frac{4}{5} x^{2} x + 2 x

= \frac{2}{9} x^{4} x + \frac{4}{5} x^{2} x + 2 x

解答补常数

= \frac{2}{9} x^{4} x + \frac{4}{5} x^{2} x + 2 x + C

t an g e n t 4 sin (x)

\frac{d}{d x} (5 x^{\frac{2}{3}})

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