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受欢迎的微积分 >

integral of 5/(x^2+8x+41)

解答

\int \frac{5}{x ^{2} + 8 x + 41} d x

解答

arctan (\frac{1}{5} (x + 4)) + C

求解步骤

\int \frac{5}{x ^{2} + 8 x + 41} 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} + 8 x + 41} d x

对

x^{2} + 8 x + 41

配方:

(x + 4)^{2} + 25

= 5 \cdot \int \frac{1}{( x + 4 ) ^{2} + 25} d x

使用换元积分法

= 5 \cdot \int \frac{1}{u ^{2} + 25} 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 + 4}{5})

化简

5 \cdot \frac{1}{5} arctan (\frac{x + 4}{5}) : arctan (\frac{1}{5} (x + 4))

= arctan (\frac{1}{5} (x + 4))

解答补常数

= arctan (\frac{1}{5} (x + 4)) + C

作图

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