integral from 0 to y^{3/2 of} x/(y^2)

解答

\int_{0}^{y^{\frac{3}{2}}} \frac{x}{y ^{2}} d x

\frac{y}{2}

求解步骤

\int_{0}^{y^{\frac{3}{2}}} \frac{x}{y ^{2}} d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{y ^{2}} \cdot \int_{0}^{y^{\frac{3}{2}}} x d x

使用幂法则

= \frac{1}{y ^{2}} [\frac{x ^{2}}{2}]_{0}^{y^{\frac{3}{2}}}

化简

\frac{1}{y ^{2}} [\frac{x ^{2}}{2}]_{0}^{y^{\frac{3}{2}}} : \frac{[ \frac{x ^{2}}{2} ] _{0}^{y^{\frac{3}{2}}}}{y ^{2}}

= \frac{[ \frac{x ^{2}}{2} ] _{0}^{y^{\frac{3}{2}}}}{y ^{2}}

带入上下限计算后得:

\frac{y ^{3}}{2}

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

化简

= \frac{y}{2}

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