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受欢迎的微积分 >

integral of (sqrt(5x^2-7))/x

解答

\int \frac{5 x ^{2} - 7}{x} d x

解答

7 (- arcsec (\frac{5}{7} x) + \frac{1}{7} (5 x^{2} - 7)) + C

求解步骤

\int \frac{5 x ^{2} - 7}{x} d x

使用三角换元法

= \int 7 tan^{2} (u) d u

提出常数:

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

= 7 \cdot \int tan^{2} (u) d u

使用三角恒等式改写

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

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

= 7 (- \int 1 d u + \int sec^{2} (u) d u)

\int 1 d u = u

\int sec^{2} (u) d u = tan (u)

= 7 (- u + tan (u))

u = arcsec (\frac{5}{7} x)

代回

= 7 (- arcsec (\frac{5}{7} x) + tan (arcsec (\frac{5}{7} x)))

化简

7 (- arcsec (\frac{5}{7} x) + tan (arcsec (\frac{5}{7} x))) : 7 (- arcsec (\frac{5}{7} x) + \frac{1}{7} (5 x^{2} - 7))

= 7 (- arcsec (\frac{5}{7} x) + \frac{1}{7} (5 x^{2} - 7))

解答补常数

= 7 (- arcsec (\frac{5}{7} x) + \frac{1}{7} (5 x^{2} - 7)) + C

作图

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