integral of (5xsqrt(x)-1/(x^3)+3)

解答

\int (5 x x - \frac{1}{x ^{3}} + 3) d x

2 x^{\frac{5}{2}} + \frac{1}{2 x ^{2}} + 3 x + C

求解步骤

\int 5 x x - \frac{1}{x ^{3}} + 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 5 x x d x - \int \frac{1}{x ^{3}} d x + \int 3 d x

\int 5 x x d x = 2 x^{\frac{5}{2}}

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

\int 3 d x = 3 x

= 2 x^{\frac{5}{2}} - (- \frac{1}{2 x ^{2}}) + 3 x

化简

= 2 x^{\frac{5}{2}} + \frac{1}{2 x ^{2}} + 3 x

解答补常数

= 2 x^{\frac{5}{2}} + \frac{1}{2 x ^{2}} + 3 x + C

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