integral of y^2(1/(y^5)+(y^4)/2)

解答

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

- \frac{1}{2 y ^{2}} + \frac{y ^{7}}{14} + C

求解步骤

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

乘开

y^{2} (\frac{1}{y ^{5}} + \frac{y ^{4}}{2}) : \frac{1}{y ^{3}} + \frac{y ^{6}}{2}

= \int \frac{1}{y ^{3}} + \frac{y ^{6}}{2} 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 \frac{1}{y ^{3}} d y + \int \frac{y ^{6}}{2} d y

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

\int \frac{y ^{6}}{2} d y = \frac{y ^{7}}{14}

= - \frac{1}{2 y ^{2}} + \frac{y ^{7}}{14}

解答补常数

= - \frac{1}{2 y ^{2}} + \frac{y ^{7}}{14} + C

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