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受欢迎的微积分 >

integral of 3/(x^2+16)

解答

\int \frac{3}{x ^{2} + 16} d x

解答

\frac{3}{4} arctan (\frac{x}{4}) + C

求解步骤

\int \frac{3}{x ^{2} + 16} d x

提出常数:

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

= 3 \cdot \int \frac{1}{x ^{2} + 16} d x

使用换元积分法

= 3 \cdot \int \frac{1}{4 ( u ^{2} + 1 )} d u

提出常数:

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

= 3 \cdot \frac{1}{4} \cdot \int \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)

= 3 \cdot \frac{1}{4} arctan (u)

u = \frac{x}{4}

代回

= 3 \cdot \frac{1}{4} arctan (\frac{x}{4})

化简

3 \cdot \frac{1}{4} arctan (\frac{x}{4}) : \frac{3}{4} arctan (\frac{x}{4})

= \frac{3}{4} arctan (\frac{x}{4})

解答补常数

= \frac{3}{4} arctan (\frac{x}{4}) + C

作图

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