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受欢迎的微积分 >

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

解答

\int \frac{5}{x ^{2} - 4 x + 9} d x

解答

5 arctan (\frac{1}{5} (x - 2)) + C

求解步骤

\int \frac{5}{x ^{2} - 4 x + 9} d x

提出常数:

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

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

对

x^{2} - 4 x + 9

配方:

(x - 2)^{2} + 5

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

使用换元积分法

= 5 \cdot \int \frac{1}{u ^{2} + 5} d u

使用换元积分法

= 5 \cdot \int \frac{1}{5 ( v ^{2} + 1 )} d v

提出常数:

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

= 5 \cdot \frac{1}{5} \cdot \int \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)

= 5 \cdot \frac{1}{5} arctan (v)

代回

= 5 \cdot \frac{1}{5} arctan (\frac{x - 2}{5})

化简

5 \cdot \frac{1}{5} arctan (\frac{x - 2}{5}) : 5 arctan (\frac{1}{5} (x - 2))

= 5 arctan (\frac{1}{5} (x - 2))

解答补常数

= 5 arctan (\frac{1}{5} (x - 2)) + C

作图

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