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受欢迎的微积分 >

integral of (sqrt(x^2-4))/(x^2)

解答

\int \frac{x ^{2} - 4}{x ^{2}} d x

解答

ln (\frac{1}{2} x^{2} - 4 + x) - \frac{x ^{2} - 4}{x} + C

求解步骤

\int \frac{x ^{2} - 4}{x ^{2}} d x

使用三角换元法

= \int tan^{2} (u) cos (u) d u

化简

tan^{2} (u) cos (u) : sin (u) tan (u)

= \int sin (u) tan (u) d u

使用三角恒等式改写

= \int \frac{1 - cos ^{2} ( u )}{cos ( u )} d u

乘开

\frac{1 - cos ^{2} ( u )}{cos ( u )} : \frac{1}{cos ( u )} - cos (u)

= \int \frac{1}{cos ( u )} - cos (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 \frac{1}{cos ( u )} d u - \int cos (u) d u

\int \frac{1}{cos ( u )} d u = ln ∣ tan (u) + sec (u) ∣

\int cos (u) d u = sin (u)

= ln ∣ tan (u) + sec (u) ∣ - sin (u)

u = arcsec (\frac{1}{2} x)

代回

= ln tan (arcsec (\frac{1}{2} x)) + sec (arcsec (\frac{1}{2} x)) - sin (arcsec (\frac{1}{2} x))

化简

ln tan (arcsec (\frac{1}{2} x)) + sec (arcsec (\frac{1}{2} x)) - sin (arcsec (\frac{1}{2} x)) : ln (\frac{1}{2} x^{2} - 4 + x) - \frac{x ^{2} - 4}{x}

= ln (\frac{1}{2} x^{2} - 4 + x) - \frac{x ^{2} - 4}{x}

解答补常数

= ln (\frac{1}{2} x^{2} - 4 + x) - \frac{x ^{2} - 4}{x} + C

作图

流行的例子

integral of (3/x+x)

\int (\frac{3}{x} + x) d x

面积 y=2sqrt(x),y=2x^3

a re a y = 2 x, y = 2 x^{3}

integral from 0 to 13/12 of 68/3 x

\int_{0}^{\frac{13}{12}} \frac{68}{3} x d x

derivative y=2x^2-9x+10

d er i v a t i v e y = 2 x^{2} - 9 x + 10

derivative of 17.5(x+1^{0.75})

\frac{d}{d x} (17.5 (x + 1)^{0.75})