integral of (5xln(x))/(sqrt(x^2-4))

解答

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

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

求解步骤

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

提出常数:

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

= 5 \cdot \int \frac{x ln ( x )}{x ^{2} - 4} d x

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

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

使用分布积分法

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

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

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

化简

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

解答补常数

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

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