Question

Evaluate the integral
∫1(x2x2−16)−−−−−−−√dx
using the substitution
x=4secθ

Answer 1

The substitution: x = 4 sec t ( or the Greek letter: theta )
dx = 4 tan t sec t

`\int { \frac{1}{ x^(2) \sqrt{ x^(2) -16} } } \, dx= \\ = \int { \frac{4 tan t sect}{16 sect \sqrt{16 sec^(2)t-16 } } } \, dt= \\ \int { (4tant)/(16sect*4tant) } \, dt = \\ = (1)/(16) \int { cos t} \, dt = (1)/(16)sin t`
From the substitution we have: cos x = 4 / x
sin² x = 1 - 16/x²
sin x =
`\frac{ \sqrt{ x^(2) -16} }{x}`
Final solution is:
`(1)/(16) * \frac{ \sqrt{ x^(2) -16} }{x} +C`