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受欢迎的微积分 >

integral of sqrt(3-2x+x^2)

解答

\int 3 - 2 x + x^{2} d x

解答

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

求解步骤

\int 3 - 2 x + x^{2} d x

对

3 - 2 x + x^{2}

配方:

(x - 1)^{2} + 2

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

使用换元积分法

= \int u^{2} + 2 d u

使用三角换元法

= \int 2 sec^{3} (v) d v

提出常数:

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

= 2 \cdot \int sec^{3} (v) d v

使用降次积分法

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

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

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

代回

= 2 \frac{sec ^{2} ( arctan ( \frac{1}{2} ( x - 1 ) ) ) sin ( arctan ( \frac{1}{2} ( x - 1 ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{2} (x - 1))) + sec (arctan (\frac{1}{2} (x - 1)))

化简

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

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

解答补常数

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

作图

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