integral from 0 to 1 of (x^2)/(1+y^2)

解答

\int_{0}^{1} \frac{x ^{2}}{1 + y ^{2}} d y

\frac{π x ^{2}}{4}

求解步骤

\int_{0}^{1} \frac{x ^{2}}{1 + 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_{0}^{1} \frac{1}{1 + y ^{2}} d y

使用积分常见定则:

\int \frac{1}{1 + y ^{2}} d y = arctan (y)

= x^{2} [arctan (y)]_{0}^{1}

带入上下限计算后得:

\frac{π}{4}

= x^{2} \frac{π}{4}

化简

= \frac{π x ^{2}}{4}

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