integral of sqrt(x^2-25)\x?

Answer 1

Calculate the integral.

∫[√(x² - 25) / x] dx
Let
x = 5secθ
dx = 5(secθ)(tanθ)dθ
x² = 25sec²θ

∫[√(25sec²θ - 25) / (5secθ)] {5(secθ)(tanθ)dθ}

= ∫[√(25tan²θ)] {(tanθ)dθ}

= ∫[5tanθ] {(tanθ)dθ}

= 5∫(tan²θ) dθ

= 5∫(sec²θ - 1) dθ

= 5tanθ - 5θ + C
__________

x = 5secθ
secθ = x/5
θ = arcsec(x/5)
tanθ = (1/5)√(x² - 25)
__________

= √(x² - 25) - 5arcsec(x/5) + C