integral from 1 to 9 of (sqrt(x^2-1))/x

解答

\int_{1}^{9} \frac{x ^{2} - 1}{x} d x

- arctan (45) + 45

十进制

7.48481 \dots

求解步骤

\int_{1}^{9} \frac{x ^{2} - 1}{x} d x

使用换元积分法

= \int_{0}^{45} \frac{u ^{2}}{u ^{2} + 1} d u

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

= \int_{0}^{45} - \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

= - \int_{0}^{45} \frac{1}{u ^{2} + 1} d u + \int_{0}^{45} 1 d u

\int_{0}^{45} \frac{1}{u ^{2} + 1} d u = arctan (45)

\int_{0}^{45} 1 d u = 45

= - arctan (45) + 45

\int \frac{5 x ^{2} - 10 x}{4 3 x ^{3}} d x

d er i v a t i v e 1 - 49 x^{2} arccos (7 x)

x \to 2 lim (7 (x) + 10)

f (x) = x^{\frac{4}{5}}

\int cos^{4} (7 x + 9) d x