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受欢迎的微积分 >

integral of sqrt(x^2-6x+13)

解答

\int x^{2} - 6 x + 13 d x

解答

4 (\frac{1}{8} (x - 3) x^{2} - 6 x + 13 + \frac{1}{2} ln (\frac{x - 3 + x ^{2} - 6 x + 13}{2})) + C

求解步骤

\int x^{2} - 6 x + 13 d x

对

x^{2} - 6 x + 13

配方:

(x - 3)^{2} + 4

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

使用换元积分法

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

使用三角换元法

= \int 4 sec^{3} (v) d v

提出常数:

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

= 4 \cdot \int sec^{3} (v) d v

使用降次积分法

= 4 (\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) ∣

= 4 (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

代回

= 4 (\frac{sec ^{2} ( arctan ( \frac{1}{2} ( x - 3 ) ) ) sin ( arctan ( \frac{1}{2} ( x - 3 ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{2} (x - 3))) + sec (arctan (\frac{1}{2} (x - 3))))

化简

4 (\frac{sec ^{2} ( arctan ( \frac{1}{2} ( x - 3 ) ) ) sin ( arctan ( \frac{1}{2} ( x - 3 ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{2} (x - 3))) + sec (arctan (\frac{1}{2} (x - 3)))) : 4 (\frac{1}{8} (x - 3) x^{2} - 6 x + 13 + \frac{1}{2} ln (\frac{x - 3 + x ^{2} - 6 x + 13}{2}))

= 4 (\frac{1}{8} (x - 3) x^{2} - 6 x + 13 + \frac{1}{2} ln (\frac{x - 3 + x ^{2} - 6 x + 13}{2}))

解答补常数

= 4 (\frac{1}{8} (x - 3) x^{2} - 6 x + 13 + \frac{1}{2} ln (\frac{x - 3 + x ^{2} - 6 x + 13}{2})) + C

作图

流行的例子

x^2(dy)/(dx)-2xy=4y^4

x^{2} \frac{d y}{d x} - 2 x y = 4 y^{4}

integral of t^2e^{-11t}

\int t^{2} e^{- 11 t} d t

limit as x approaches 2 of 3^{x+1}

x \to 2 lim (3^{x + 1})

limit as x approaches c of f(c)

x \to c lim (f (c))

integral of 1arcsin(x)

\int 1 arcsin (x) d x