antiderivative and trig substitution

Question

if you were looking for the antiderivative of f(x)=1/[x(x²-1)^3/2], which trigonometric substitution would be appropriate, a sine substitution, a tangent substitution or a secant substitution? justify your choice

Answer 1

The best substitution for this problem would be to let x= secθ due to the form x²-1 which when using x=secθ gives sec²θ -1 which is the tan²θ. The function becomes f(θ) = 1/secθtan³θ which simplifies to cos⁴θ/sin³θ. This can be solved using u substitution techniques.