integral from 1/x to x of (x^2)/(y^2)

解答

\int_{\frac{1}{x}}^{x} \frac{x ^{2}}{y ^{2}} d y

x^{2} (- \frac{1}{x} + x)

求解步骤

\int_{\frac{1}{x}}^{x} \frac{x ^{2}}{y ^{2}} d y

提出常数:

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

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

使用幂法则

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

带入上下限计算后得:

- \frac{1}{x} + x

= x^{2} (- \frac{1}{x} + x)

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