integral from 0 to 1 of 8cos(x^2)

解答

\int_{0}^{1} 8 cos (x^{2}) d x

8 (\frac{π}{2} C (\frac{2}{π} \cdot 1) - \frac{π}{2} C (\frac{2}{π} \cdot 0))

求解步骤

\int_{0}^{1} 8 cos (x^{2}) d x

提出常数:

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

= 8 \cdot \int_{0}^{1} cos (x^{2}) d x

这是非初等函数积分:

\int cos (x^{2}) d x = \frac{π}{2} C (\frac{2}{π} x)

= 8 [\frac{π}{2} C (\frac{2}{π} x)]_{0}^{1}

带入上下限计算后得:

\frac{π}{2} C (\frac{2}{π} \cdot 1) - \frac{π}{2} C (\frac{2}{π} \cdot 0)

= 8 (\frac{π}{2} C (\frac{2}{π} \cdot 1) - \frac{π}{2} C (\frac{2}{π} \cdot 0))

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