integral of (x^2)/(sqrt(x^2-7))

解答

\int \frac{x ^{2}}{x ^{2} - 7} d x

7 (\frac{1}{14} x x^{2} - 7 + \frac{1}{2} ln \frac{1}{7} (x^{2} - 7) + \frac{1}{7} x) + C

求解步骤

\int \frac{x ^{2}}{x ^{2} - 7} d x

使用三角换元法

= \int 7 sec^{3} (u) d u

提出常数:

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

= 7 \cdot \int sec^{3} (u) d u

使用降次积分法

= 7 (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} \cdot \int sec (u) d u)

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

= 7 (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

u = arcsec (\frac{1}{7} x)

代回

= 7 \frac{sec ^{2} ( arcsec ( \frac{1}{7} x ) ) sin ( arcsec ( \frac{1}{7} x ) )}{2} + \frac{1}{2} ln tan (arcsec (\frac{1}{7} x)) + sec (arcsec (\frac{1}{7} x))

化简

7 \frac{sec ^{2} ( arcsec ( \frac{1}{7} x ) ) sin ( arcsec ( \frac{1}{7} x ) )}{2} + \frac{1}{2} ln tan (arcsec (\frac{1}{7} x)) + sec (arcsec (\frac{1}{7} x)) : 7 (\frac{1}{14} x x^{2} - 7 + \frac{1}{2} ln \frac{1}{7} (x^{2} - 7) + \frac{1}{7} x)

= 7 (\frac{1}{14} x x^{2} - 7 + \frac{1}{2} ln \frac{1}{7} (x^{2} - 7) + \frac{1}{7} x)

解答补常数

= 7 (\frac{1}{14} x x^{2} - 7 + \frac{1}{2} ln \frac{1}{7} (x^{2} - 7) + \frac{1}{7} x) + C

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