integral of (x^5-5^x+\sqrt[5]{x}-5/x)

解答

\int (x^{5} - 5^{x} + 5 x - \frac{5}{x}) d x

\frac{x ^{6}}{6} - \frac{5 ^{x}}{ln ( 5 )} + \frac{5}{6} x^{\frac{6}{5}} - 5 ln ∣ x ∣ + C

求解步骤

\int x^{5} - 5^{x} + 5 x - \frac{5}{x} d x

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

= \int x^{5} d x - \int 5^{x} d x + \int 5 x d x - \int \frac{5}{x} d x

\int x^{5} d x = \frac{x ^{6}}{6}

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

\int 5 x d x = \frac{5}{6} x^{\frac{6}{5}}

\int \frac{5}{x} d x = 5 ln ∣ x ∣

= \frac{x ^{6}}{6} - \frac{5 ^{x}}{ln ( 5 )} + \frac{5}{6} x^{\frac{6}{5}} - 5 ln ∣ x ∣

解答补常数

= \frac{x ^{6}}{6} - \frac{5 ^{x}}{ln ( 5 )} + \frac{5}{6} x^{\frac{6}{5}} - 5 ln ∣ x ∣ + C

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