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受欢迎的微积分 >

integral of 1/(2x^2+3x+5)

解答

\int \frac{1}{2 x ^{2} + 3 x + 5} d x

解答

\frac{2}{31} arctan (\frac{4 x + 3}{31}) + C

求解步骤

\int \frac{1}{2 x ^{2} + 3 x + 5} d x

对

2 x^{2} + 3 x + 5

配方:

2 (x + \frac{3}{4})^{2} + \frac{31}{8}

= \int \frac{1}{2 ( x + \frac{3}{4} ) ^{2} + \frac{31}{8}} d x

使用换元积分法

= \int \frac{8}{16 u ^{2} + 31} d u

提出常数:

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

= 8 \cdot \int \frac{1}{16 u ^{2} + 31} d u

使用换元积分法

= 8 \cdot \int \frac{1}{4 31 ( v ^{2} + 1 )} d v

提出常数:

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

= 8 \cdot \frac{1}{4 31} \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)

= 8 \cdot \frac{1}{4 31} arctan (v)

代回

= 8 \cdot \frac{1}{4 31} arctan (\frac{4}{31} (x + \frac{3}{4}))

化简

8 \cdot \frac{1}{4 31} arctan (\frac{4}{31} (x + \frac{3}{4})) : \frac{2}{31} arctan (\frac{4 x + 3}{31})

= \frac{2}{31} arctan (\frac{4 x + 3}{31})

解答补常数

= \frac{2}{31} arctan (\frac{4 x + 3}{31}) + C

作图

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