integral of (y^{5/2}-5y^{4/3}-2y^{1/4}-sqrt(y))

解答

\int (y^{\frac{5}{2}} - 5 y^{\frac{4}{3}} - 2 y^{\frac{1}{4}} - y) d y

\frac{2}{7} y^{\frac{7}{2}} - \frac{15}{7} y^{\frac{7}{3}} - \frac{8}{5} y^{\frac{5}{4}} - \frac{2}{3} y^{\frac{3}{2}} + C

求解步骤

\int y^{\frac{5}{2}} - 5 y^{\frac{4}{3}} - 2 y^{\frac{1}{4}} - y d y

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 y^{\frac{5}{2}} d y - \int 5 y^{\frac{4}{3}} d y - \int 2 y^{\frac{1}{4}} d y - \int y d y

\int y^{\frac{5}{2}} d y = \frac{2}{7} y^{\frac{7}{2}}

\int 5 y^{\frac{4}{3}} d y = \frac{15}{7} y^{\frac{7}{3}}

\int 2 y^{\frac{1}{4}} d y = \frac{8}{5} y^{\frac{5}{4}}

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

= \frac{2}{7} y^{\frac{7}{2}} - \frac{15}{7} y^{\frac{7}{3}} - \frac{8}{5} y^{\frac{5}{4}} - \frac{2}{3} y^{\frac{3}{2}}

解答补常数

= \frac{2}{7} y^{\frac{7}{2}} - \frac{15}{7} y^{\frac{7}{3}} - \frac{8}{5} y^{\frac{5}{4}} - \frac{2}{3} y^{\frac{3}{2}} + C

y^{^{'}} = y (y + 1)

in v erse l a pl a ce \frac{2}{s ^{2} + 9}

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