integral from 0 to 2 of 2/(sqrt(6-x^2))

解答

\int_{0}^{2} \frac{2}{6 - x ^{2}} d x

2 arcsin (\frac{2}{3})

十进制

1.91063 \dots

求解步骤

\int_{0}^{2} \frac{2}{6 - x ^{2}} d x

提出常数:

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

= 2 \cdot \int_{0}^{2} \frac{1}{6 - x ^{2}} d x

使用三角换元法

= 2 \cdot \int_{0}^{a r c s i n (\frac{2}{3})} 1 d u

常数积分:

\int a d x = a x

= 2 [1 \cdot u]_{0}^{a r c s i n (\frac{2}{3})}

化简

= 2 [u]_{0}^{a r c s i n (\frac{2}{3})}

带入上下限计算后得:

arcsin (\frac{2}{3})

= 2 arcsin (\frac{2}{3})

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