evaluate the integral below

Question

evaluate the integral: (sqrt(49 - x^2)) / (12x^2)

the trig substitution for this is: 7sin(theta)

1. Which integral do you obtain after substituting for x and simplifying?

2. What is the value of the above integral in terms of θ ?

3. What is the value of the original integral in terms of x ?

please number each of the answers!

Answer 1

For x=7sinθ dx= 7cosθdθ and the integral after substitution becomes ∫√(49-49sin²θ)7cosθdθ/12(49)sin²θ. This can be simplified to 49∫cosθcosθdθ/12(49)sin²θ which becomes ∫cos²θdθ/12sin²θ. Using the trig identity for cos²θ the integral becomes ∫(1-sin²θ)dθ/12sin²θ which gives ∫(csc²θ -1dθ/12 (1). This integrates to (-ctnθ - θ)/12 + C (2). Since x=7sinθ θ will equal sin^-1(x/7) and ctnθ =√(49 - x²)/x. Substituting these into the θ equation gives the integral is (1/12)( -√(49 - x²)/x - sin^-1(x/7) ) + C