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受欢迎的微积分 >

integral from 0 to 1/6 of 3/(sqrt(1-9x^2))

解答

\int_{0}^{\frac{1}{6}} \frac{3}{1 - 9 x ^{2}} d x

解答

\frac{π}{6}

+1

十进制

0.52359 \dots

求解步骤

\int_{0}^{\frac{1}{6}} \frac{3}{1 - 9 x ^{2}} d x

提出常数:

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

= 3 \cdot \int_{0}^{\frac{1}{6}} \frac{1}{1 - 9 x ^{2}} d x

使用三角换元法

= 3 \cdot \int_{0}^{\frac{π}{6}} \frac{1}{3} d u

提出常数:

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

= 3 \cdot \frac{1}{3} \cdot \int_{0}^{\frac{π}{6}} 1 d u

常数积分:

\int a d x = a x

= 3 \cdot \frac{1}{3} [1 \cdot u]_{0}^{\frac{π}{6}}

化简

3 \cdot \frac{1}{3} [1 \cdot u]_{0}^{\frac{π}{6}} : [u]_{0}^{\frac{π}{6}}

= [u]_{0}^{\frac{π}{6}}

带入上下限计算后得:

\frac{π}{6}

= \frac{π}{6}

作图

流行的例子

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y^{^{''}} - 2 y^{^{'}} + 1 y = 0, y (0) = 8, y^{^{'}} (0) = 2

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