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integral of (4x^2-5)^{1/2}

解答

\int (4 x^{2} - 5)^{\frac{1}{2}} d x

解答

\frac{1}{4} (2 x (4 x^{2} - 5)^{\frac{1}{2}} - 5 (ln 2 x + 5^{\frac{1}{2}} (\frac{4 x ^{2} - 5}{5})^{\frac{1}{2}} - \frac{ln ( 5 )}{2})) + C

求解步骤

\int (4 x^{2} - 5)^{\frac{1}{2}} d x

使用三角换元法

= \int \frac{5}{2} tan^{2} (u) sec (u) d u

提出常数:

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

= \frac{5}{2} \cdot \int tan^{2} (u) sec (u) d u

使用三角恒等式改写

= \frac{5}{2} \cdot \int (- 1 + sec^{2} (u)) sec (u) d u

乘开

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

= \frac{5}{2} \cdot \int - sec (u) + sec^{3} (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

= \frac{5}{2} (- \int sec (u) d u + \int sec^{3} (u) d u)

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

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

= \frac{5}{2} (- ln ∣ tan (u) + sec (u) ∣ + \frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

u = arcsec (\frac{2}{5 ^{\frac{1}{2}}} x)

代回

= \frac{5}{2} (- ln tan (arcsec (\frac{2}{5 ^{\frac{1}{2}}} x)) + sec (arcsec (\frac{2}{5 ^{\frac{1}{2}}} x)) + \frac{1}{2} sec (arcsec (\frac{2}{5 ^{\frac{1}{2}}} x)) tan (arcsec (\frac{2}{5 ^{\frac{1}{2}}} x)) + \frac{1}{2} ln tan (arcsec (\frac{2}{5 ^{\frac{1}{2}}} x)) + sec (arcsec (\frac{2}{5 ^{\frac{1}{2}}} x)))

化简

\frac{5}{2} (- ln tan (arcsec (\frac{2}{5 ^{\frac{1}{2}}} x)) + sec (arcsec (\frac{2}{5 ^{\frac{1}{2}}} x)) + \frac{1}{2} sec (arcsec (\frac{2}{5 ^{\frac{1}{2}}} x)) tan (arcsec (\frac{2}{5 ^{\frac{1}{2}}} x)) + \frac{1}{2} ln tan (arcsec (\frac{2}{5 ^{\frac{1}{2}}} x)) + sec (arcsec (\frac{2}{5 ^{\frac{1}{2}}} x))) : \frac{1}{4} (2 x (4 x^{2} - 5)^{\frac{1}{2}} - 5 (ln 2 x + 5^{\frac{1}{2}} (\frac{4 x ^{2} - 5}{5})^{\frac{1}{2}} - \frac{ln ( 5 )}{2}))

= \frac{1}{4} (2 x (4 x^{2} - 5)^{\frac{1}{2}} - 5 (ln 2 x + 5^{\frac{1}{2}} (\frac{4 x ^{2} - 5}{5})^{\frac{1}{2}} - \frac{ln ( 5 )}{2}))

解答补常数

= \frac{1}{4} (2 x (4 x^{2} - 5)^{\frac{1}{2}} - 5 (ln 2 x + 5^{\frac{1}{2}} (\frac{4 x ^{2} - 5}{5})^{\frac{1}{2}} - \frac{ln ( 5 )}{2})) + C

作图

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