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受欢迎的微积分 >

integral from-2 to 2 of 8/(4+x^2)

解答

\int_{- 2}^{2} \frac{8}{4 + x ^{2}} d x

解答

2 π

+1

十进制

6.28318 \dots

求解步骤

\int_{- 2}^{2} \frac{8}{4 + x ^{2}} d x

提出常数:

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

= 8 \cdot \int_{- 2}^{2} \frac{1}{4 + x ^{2}} d x

使用换元积分法

= 8 \cdot \int_{- 1}^{1} \frac{1}{2 ( u ^{2} + 1 )} d u

提出常数:

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

= 8 \cdot \frac{1}{2} \cdot \int_{- 1}^{1} \frac{1}{u ^{2} + 1} d u

使用积分常见定则:

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

= 8 \cdot \frac{1}{2} [arctan (u)]_{- 1}^{1}

化简

8 \cdot \frac{1}{2} [arctan (u)]_{- 1}^{1} : 4 [arctan (u)]_{- 1}^{1}

= 4 [arctan (u)]_{- 1}^{1}

带入上下限计算后得:

\frac{π}{2}

= 4 \cdot \frac{π}{2}

化简

= 2 π

作图

流行的例子

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\int_{2}^{4} (x^{2} + 2 x) d x

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\int_{- 1}^{0} - x cos (nπ x) d x

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\int_{2}^{6} 2 x^{2} + 4 d x

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\int_{- 2}^{4} x^{2} d x

integral from 0 to pi/4 of sin(θ)

\int_{0}^{\frac{π}{4}} sin (θ) d θ