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受欢迎的微积分 >

integral of 6/(x^2+4)

解答

\int \frac{6}{x ^{2} + 4} d x

解答

3 arctan (\frac{x}{2}) + C

求解步骤

\int \frac{6}{x ^{2} + 4} d x

提出常数:

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

= 6 \cdot \int \frac{1}{x ^{2} + 4} d x

使用换元积分法

= 6 \cdot \int \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

= 6 \cdot \frac{1}{2} \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)

= 6 \cdot \frac{1}{2} arctan (u)

u = \frac{x}{2}

代回

= 6 \cdot \frac{1}{2} arctan (\frac{x}{2})

化简

6 \cdot \frac{1}{2} arctan (\frac{x}{2}) : 3 arctan (\frac{x}{2})

= 3 arctan (\frac{x}{2})

解答补常数

= 3 arctan (\frac{x}{2}) + C

作图

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