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受欢迎的微积分 >

integral from 0 to 12 of 5/(1+0.2t^2)

解答

\int_{0}^{12} \frac{5}{1 + 0.2 t ^{2}} d t

解答

15.50232 \dots

求解步骤

\int_{0}^{12} \frac{5}{1 + 0.2 t ^{2}} d t

提出常数:

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

= 5 \cdot \int_{0}^{12} \frac{1}{1 + 0.2 t ^{2}} d t

使用换元积分法

= 5 \cdot \int_{0}^{5.36656 \dots} \frac{2.23606 \dots}{1 + u ^{2}} d u

提出常数:

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

= 5 \cdot 2.23606 \dots \cdot \int_{0}^{5.36656 \dots} \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)

= 5 \cdot 2.23606 \dots [arctan (u)]_{0}^{5.36656 \dots}

化简

= 11.18033 \dots [arctan (u)]_{0}^{5.36656 \dots}

带入上下限计算后得:

1.38657 \dots

= 11.18033 \dots \cdot 1.38657 \dots

化简

= 15.50232 \dots

作图

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