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受欢迎的微积分 >

integral from 0 to 1 of (24)/(5+3x^2)

解答

\int_{0}^{1} \frac{24}{5 + 3 x ^{2}} d x

解答

\frac{8 3}{5} arctan (\frac{3}{5})

+1

十进制

4.08403 \dots

求解步骤

\int_{0}^{1} \frac{24}{5 + 3 x ^{2}} d x

提出常数:

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

= 24 \cdot \int_{0}^{1} \frac{1}{5 + 3 x ^{2}} d x

使用换元积分法

= 24 \cdot \int_{0}^{\frac{3}{5}} \frac{1}{15 ( 1 + u ^{2} )} d u

提出常数:

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

= 24 \cdot \frac{1}{15} \cdot \int_{0}^{\frac{3}{5}} \frac{1}{1 + u ^{2}} d u

使用积分常见定则:

\int \frac{1}{1 + u ^{2}} d u = arctan (u)

= 24 \cdot \frac{1}{15} [arctan (u)]_{0}^{\frac{3}{5}}

化简

24 \cdot \frac{1}{15} [arctan (u)]_{0}^{\frac{3}{5}} : \frac{8 3}{5} [arctan (u)]_{0}^{\frac{3}{5}}

= \frac{8 3}{5} [arctan (u)]_{0}^{\frac{3}{5}}

带入上下限计算后得:

arctan (\frac{3}{5})

= \frac{8 3}{5} arctan (\frac{3}{5})

作图

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