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受欢迎的微积分 >

integral from 2 to 5 of 4/(9+(x-2)^2)

解答

\int_{2}^{5} \frac{4}{9 + ( x - 2 ) ^{2}} d x

解答

\frac{π}{3}

+1

十进制

1.04719 \dots

求解步骤

\int_{2}^{5} \frac{4}{9 + ( x - 2 ) ^{2}} d x

提出常数:

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

= 4 \cdot \int_{2}^{5} \frac{1}{9 + ( x - 2 ) ^{2}} d x

使用换元积分法

= 4 \cdot \int_{0}^{3} \frac{1}{9 + u ^{2}} d u

使用换元积分法

= 4 \cdot \int_{0}^{1} \frac{1}{3 ( v ^{2} + 1 )} d v

提出常数:

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

= 4 \cdot \frac{1}{3} \cdot \int_{0}^{1} \frac{1}{v ^{2} + 1} d v

使用积分常见定则:

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

= 4 \cdot \frac{1}{3} [arctan (v)]_{0}^{1}

化简

4 \cdot \frac{1}{3} [arctan (v)]_{0}^{1} : \frac{4}{3} [arctan (v)]_{0}^{1}

= \frac{4}{3} [arctan (v)]_{0}^{1}

带入上下限计算后得:

\frac{π}{4}

= \frac{4}{3} \cdot \frac{π}{4}

化简

= \frac{π}{3}

作图

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