integral from 1 to x of 2(1-1/(y^2))

解答

\int_{1}^{x} 2 (1 - \frac{1}{y ^{2}}) d y

2 (x + \frac{1}{x} - 2)

求解步骤

\int_{1}^{x} 2 (1 - \frac{1}{y ^{2}}) d y

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int_{1}^{x} 1 - \frac{1}{y ^{2}} d y

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

= 2 (\int_{1}^{x} 1 d y - \int_{1}^{x} \frac{1}{y ^{2}} d y)

\int_{1}^{x} 1 d y = x - 1

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

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

化简

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

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