integral of (x^{8.3}-9x^6+3x^{-4}+x^{-3})

解答

\int (x^{8.3} - 9 x^{6} + 3 x^{- 4} + x^{- 3}) d x

0.10752 \dots x^{9.3} - 1.28571 \dots x^{7} - \frac{1}{x ^{3}} - \frac{0.5}{x ^{2}} + C

求解步骤

\int x^{8.3} - 9 x^{6} + 3 x^{- 4} + x^{- 3} d x

化简

x^{8.3} - 9 x^{6} + 3 x^{- 4} + x^{- 3} : x^{8.3} - 9 x^{6} + \frac{3}{x ^{4}} + \frac{1}{x ^{3}}

= \int x^{8.3} - 9 x^{6} + \frac{3}{x ^{4}} + \frac{1}{x ^{3}} 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^{8.3} d x - \int 9 x^{6} d x + \int \frac{3}{x ^{4}} d x + \int \frac{1}{x ^{3}} d x

\int x^{8.3} d x = 0.10752 \dots x^{9.3}

\int 9 x^{6} d x = \frac{9 x ^{7}}{7}

\int \frac{3}{x ^{4}} d x = - \frac{1}{x ^{3}}

\int \frac{1}{x ^{3}} d x = - \frac{1}{2 x ^{2}}

= 0.10752 \dots x^{9.3} - \frac{9 x ^{7}}{7} - \frac{1}{x ^{3}} - \frac{1}{2 x ^{2}}

化简

= 0.10752 \dots x^{9.3} - 1.28571 \dots x^{7} - \frac{1}{x ^{3}} - \frac{0.5}{x ^{2}}

解答补常数

= 0.10752 \dots x^{9.3} - 1.28571 \dots x^{7} - \frac{1}{x ^{3}} - \frac{0.5}{x ^{2}} + C

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