integral of (5x^{1/4}-5x^{-6}+1)

解答

\int (5 x^{\frac{1}{4}} - 5 x^{- 6} + 1) d x

4 x^{\frac{5}{4}} + \frac{1}{x ^{5}} + x + C

求解步骤

\int 5 x^{\frac{1}{4}} - 5 x^{- 6} + 1 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 5 x^{\frac{1}{4}} d x - \int 5 x^{- 6} d x + \int 1 d x

\int 5 x^{\frac{1}{4}} d x = 4 x^{\frac{5}{4}}

\int 5 x^{- 6} d x = - \frac{1}{x ^{5}}

\int 1 d x = x

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

化简

= 4 x^{\frac{5}{4}} + \frac{1}{x ^{5}} + x

解答补常数

= 4 x^{\frac{5}{4}} + \frac{1}{x ^{5}} + x + C

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