integral of (sqrt(x^2+81))/(8x^2)

解答

\int \frac{x ^{2} + 81}{8 x ^{2}} d x

\frac{1}{8} (- \frac{x ^{2} + 81}{x} + ln (\frac{1}{9} x + 81 + x^{2})) + C

求解步骤

\int \frac{x ^{2} + 81}{8 x ^{2}} d x

提出常数:

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

= \frac{1}{8} \cdot \int \frac{x ^{2} + 81}{x ^{2}} d x

使用分布积分法

= \frac{1}{8} (- \frac{x ^{2} + 81}{x} - \int - \frac{1}{x ^{2} + 81} d x)

\int - \frac{1}{x ^{2} + 81} d x = - ln (\frac{1}{9} x + 81 + x^{2})

= \frac{1}{8} (- \frac{x ^{2} + 81}{x} - (- ln (\frac{1}{9} x + 81 + x^{2})))

化简

= \frac{1}{8} (- \frac{x ^{2} + 81}{x} + ln (\frac{1}{9} x + 81 + x^{2}))

解答补常数

= \frac{1}{8} (- \frac{x ^{2} + 81}{x} + ln (\frac{1}{9} x + 81 + x^{2})) + C

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