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受欢迎的微积分 >

integral of 1/(sqrt(1+x+x^2))

解答

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

解答

ln \frac{2}{3} x + \frac{1}{3} + \frac{1}{3} (4 x^{2} + 4 x + 4) + C

求解步骤

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

对

1 + x + x^{2}

配方:

(x + \frac{1}{2})^{2} + \frac{3}{4}

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

使用换元积分法

= \int \frac{2}{4 u ^{2} + 3} d u

提出常数:

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

= 2 \cdot \int \frac{1}{4 u ^{2} + 3} d u

使用三角换元法

= 2 \cdot \int \frac{sec ( v )}{2} d v

提出常数:

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

= 2 \cdot \frac{1}{2} \cdot \int sec (v) d v

使用积分常见定则:

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

= 2 \cdot \frac{1}{2} ln ∣ tan (v) + sec (v) ∣

代回

= 2 \cdot \frac{1}{2} ln tan (arctan (\frac{2}{3} (x + \frac{1}{2}))) + sec (arctan (\frac{2}{3} (x + \frac{1}{2})))

化简

2 \cdot \frac{1}{2} ln tan (arctan (\frac{2}{3} (x + \frac{1}{2}))) + sec (arctan (\frac{2}{3} (x + \frac{1}{2}))) : ln \frac{2}{3} x + \frac{1}{3} + \frac{1}{3} (4 x^{2} + 4 x + 4)

= ln \frac{2}{3} x + \frac{1}{3} + \frac{1}{3} (4 x^{2} + 4 x + 4)

解答补常数

= ln \frac{2}{3} x + \frac{1}{3} + \frac{1}{3} (4 x^{2} + 4 x + 4) + C

作图

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