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受欢迎的微积分 >

integral of sqrt(x(x-1))

解答

\int x (x - 1) d x

解答

\frac{1}{8} (4 x x (x - 1) - 2 x (x - 1) - ln 2 x - 1 + 2 x (x - 1)) + C

求解步骤

\int x (x - 1) d x

乘开

x (x - 1) : x^{2} - x

= \int x^{2} - x d x

对

x^{2} - x

配方:

(x - \frac{1}{2})^{2} - \frac{1}{4}

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

使用换元积分法

= \int \frac{4 u ^{2} - 1}{2} d u

提出常数:

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

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

使用三角换元法

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

提出常数:

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

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

使用三角恒等式改写

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

乘开

(- 1 + sec^{2} (v)) sec (v) : - sec (v) + sec^{3} (v)

= \frac{1}{2} \cdot \frac{1}{2} \cdot \int - sec (v) + sec^{3} (v) d v

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} \cdot \frac{1}{2} (- \int sec (v) d v + \int sec^{3} (v) d v)

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

\int sec^{3} (v) d v = \frac{1}{2} sec (v) tan (v) + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣

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

代回

= \frac{1}{2} \cdot \frac{1}{2} (- ln tan (arcsec (2 (x - \frac{1}{2}))) + sec (arcsec (2 (x - \frac{1}{2}))) + \frac{1}{2} sec (arcsec (2 (x - \frac{1}{2}))) tan (arcsec (2 (x - \frac{1}{2}))) + \frac{1}{2} ln tan (arcsec (2 (x - \frac{1}{2}))) + sec (arcsec (2 (x - \frac{1}{2}))))

化简

\frac{1}{2} \cdot \frac{1}{2} (- ln tan (arcsec (2 (x - \frac{1}{2}))) + sec (arcsec (2 (x - \frac{1}{2}))) + \frac{1}{2} sec (arcsec (2 (x - \frac{1}{2}))) tan (arcsec (2 (x - \frac{1}{2}))) + \frac{1}{2} ln tan (arcsec (2 (x - \frac{1}{2}))) + sec (arcsec (2 (x - \frac{1}{2})))) : \frac{1}{8} (4 x x (x - 1) - 2 x (x - 1) - ln 2 x - 1 + 2 x (x - 1))

= \frac{1}{8} (4 x x (x - 1) - 2 x (x - 1) - ln 2 x - 1 + 2 x (x - 1))

解答补常数

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

作图

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