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受欢迎的微积分 >

integral of cos(8x^2+pi)

解答

\int cos (8 x^{2} + π) d x

解答

- \frac{π}{4} C (\frac{2 \cdot 2 x}{π}) + C

求解步骤

\int cos (8 x^{2} + π) d x

使用三角恒等式改写

= \int - cos (8 x^{2}) d x

提出常数:

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

= - \int cos (8 x^{2}) d x

使用换元积分法

= - \int \frac{1}{2} cos (2 u^{2}) d u

提出常数:

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

= - \frac{1}{2} \cdot \int cos (2 u^{2}) d u

这是非初等函数积分:

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

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

u = 2 x

代回

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

化简

- \frac{1}{2} \cdot \frac{π}{2} C (\frac{2 \cdot 2 x}{π}) : - \frac{π}{4} C (\frac{2 \cdot 2 x}{π})

= - \frac{π}{4} C (\frac{2 \cdot 2 x}{π})

解答补常数

= - \frac{π}{4} C (\frac{2 \cdot 2 x}{π}) + C

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