integral from 1 to 2 of (5sqrt(x^2-1))/x

解答

\int_{1}^{2} \frac{5 x ^{2} - 1}{x} d x

5 (- \frac{π}{3} + 3)

十进制

3.42426 \dots

求解步骤

\int_{1}^{2} \frac{5 x ^{2} - 1}{x} d x

提出常数:

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

= 5 \cdot \int_{1}^{2} \frac{x ^{2} - 1}{x} d x

使用换元积分法

= 5 \cdot \int_{0}^{3} \frac{u ^{2}}{u ^{2} + 1} d u

\frac{u ^{2}}{u ^{2} + 1} = - \frac{1}{u ^{2} + 1} + 1

= 5 \cdot \int_{0}^{3} - \frac{1}{u ^{2} + 1} + 1 d u

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

= 5 (- \int_{0}^{3} \frac{1}{u ^{2} + 1} d u + \int_{0}^{3} 1 d u)

\int_{0}^{3} \frac{1}{u ^{2} + 1} d u = \frac{π}{3}

\int_{0}^{3} 1 d u = 3

= 5 (- \frac{π}{3} + 3)

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