integral of 2x+1/y-y/(x^2)

解答

\int 2 x + \frac{1}{y} - \frac{y}{x ^{2}} d x

\frac{x}{y} + \frac{y}{x} + x^{2} + C

求解步骤

\int 2 x + \frac{1}{y} - \frac{y}{x ^{2}} 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 \frac{1}{y} d x - \int \frac{y}{x ^{2}} d x + \int 2 x d x

\int \frac{1}{y} d x = \frac{x}{y}

\int \frac{y}{x ^{2}} d x = - \frac{y}{x}

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

= \frac{x}{y} - (- \frac{y}{x}) + x^{2}

化简

= \frac{x}{y} + \frac{y}{x} + x^{2}

解答补常数

= \frac{x}{y} + \frac{y}{x} + x^{2} + C

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