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受欢迎的微积分 >

integral of 9/(9+x^2)

解答

\int \frac{9}{9 + x ^{2}} d x

解答

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

求解步骤

\int \frac{9}{9 + x ^{2}} d x

提出常数:

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

= 9 \cdot \int \frac{1}{9 + x ^{2}} d x

使用换元积分法

= 9 \cdot \int \frac{1}{3 ( u ^{2} + 1 )} d u

提出常数:

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

= 9 \cdot \frac{1}{3} \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)

= 9 \cdot \frac{1}{3} arctan (u)

u = \frac{x}{3}

代回

= 9 \cdot \frac{1}{3} arctan (\frac{x}{3})

化简

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

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

解答补常数

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

作图

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