integral from 0 to x of sqrt(64-t^2)

解答

\int_{0}^{x} 64 - t^{2} d t

32 (arcsin (\frac{x}{8}) + \frac{1}{2} sin (2 arcsin (\frac{x}{8})))

求解步骤

\int_{0}^{x} 64 - t^{2} d t

使用三角换元法

= \int_{0}^{a r c s i n (\frac{x}{8})} 64 cos^{2} (u) d u

提出常数:

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

= 64 \cdot \int_{0}^{a r c s i n (\frac{x}{8})} cos^{2} (u) d u

使用三角恒等式改写

= 64 \cdot \int_{0}^{a r c s i n (\frac{x}{8})} \frac{1 + cos ( 2 u )}{2} d u

提出常数:

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

= 64 \cdot \frac{1}{2} \cdot \int_{0}^{a r c s i n (\frac{x}{8})} 1 + cos (2 u) d u

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 64 \cdot \frac{1}{2} (\int_{0}^{a r c s i n (\frac{x}{8})} 1 d u + \int_{0}^{a r c s i n (\frac{x}{8})} cos (2 u) d u)

\int_{0}^{a r c s i n (\frac{x}{8})} 1 d u = arcsin (\frac{x}{8})

\int_{0}^{a r c s i n (\frac{x}{8})} cos (2 u) d u = \frac{1}{2} sin (2 arcsin (\frac{x}{8}))

= 64 \cdot \frac{1}{2} (arcsin (\frac{x}{8}) + \frac{1}{2} sin (2 arcsin (\frac{x}{8})))

化简

64 \cdot \frac{1}{2} (arcsin (\frac{x}{8}) + \frac{1}{2} sin (2 arcsin (\frac{x}{8}))) : 32 (arcsin (\frac{x}{8}) + \frac{1}{2} sin (2 arcsin (\frac{x}{8})))

= 32 (arcsin (\frac{x}{8}) + \frac{1}{2} sin (2 arcsin (\frac{x}{8})))

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