integral of sqrt((x^2-1)/(x^2))

解答

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

- arctan (- 1 + x^{2}) + - 1 + x^{2} + C

求解步骤

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

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

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

化简

- \frac{1}{x ^{2}} + 1 : \frac{- 1 + x ^{2}}{x}

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

使用换元积分法

= \int \frac{u ^{2}}{u ^{2} + 1} d u

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

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

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

\int 1 d u = u

= - arctan (u) + u

u = - 1 + x^{2}

代回

= - arctan (- 1 + x^{2}) + - 1 + x^{2}

解答补常数

= - arctan (- 1 + x^{2}) + - 1 + x^{2} + C

\int - \frac{1}{3 y} d y

\int e^{\frac{- 1}{2} x} d x

\int \frac{2 - sin ( x )}{2 x + cos ( x )} d x

\int (4 x + 19)^{- 2} d x

\int 20 e^{4 t} 7 + e^{4 t} d t