integral of (sqrt(9-x^2))/(x^2)

解答

\int \frac{9 - x ^{2}}{x ^{2}} d x

- arcsin (\frac{1}{3} x) - \frac{9 - x ^{2}}{x} + C

求解步骤

\int \frac{9 - x ^{2}}{x ^{2}} d x

使用三角换元法

= \int cot^{2} (u) d u

使用三角恒等式改写

= \int - 1 + csc^{2} (u) 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 1 d u + \int csc^{2} (u) d u

\int 1 d u = u

\int csc^{2} (u) d u = - cot (u)

= - u - cot (u)

u = arcsin (\frac{1}{3} x)

代回

= - arcsin (\frac{1}{3} x) - cot (arcsin (\frac{1}{3} x))

化简

- arcsin (\frac{1}{3} x) - cot (arcsin (\frac{1}{3} x)) : - arcsin (\frac{1}{3} x) - \frac{9 - x ^{2}}{x}

= - arcsin (\frac{1}{3} x) - \frac{9 - x ^{2}}{x}

解答补常数

= - arcsin (\frac{1}{3} x) - \frac{9 - x ^{2}}{x} + C

y^{^{'}} = t^{0.5}, y (0) = 0

\frac{d}{d x} (5 x e^{3 x})

\int_{0}^{\frac{π}{3}} 8 sec^{3} (x) d x

t an g e n t f (x) = 7 x + 2, (2, 4)

x y^{^{'}} + y = e^{x}, y (1) = 1