important please evaluate the integral

Question

evaluate the integral: -16 / x^2(sqrt(x^2+100))

the trig substitution for this is: 10tan(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 ?

when answering this, i would appreciate it if you could please number each of your FINAL answers with #1, #2, and #3, as i get easily confused.

Answer 1

Using the trig substitution x= 10 tanθ and dx = 10 sec²θ in the original integral along with the identity 1+tan²θ =sec²θ it becomes (-16/100)∫sec²θ/tan²θsecθ. This simplifies to (-16/100)∫secθdθ/tan²θ which can be written as (-16/100)∫cos²θdθ/cosθsin²θ which now becomes(-16/100)∫cosθdθ/sin²θ. (1) Using a u substitution for sinθ where du becomes cosθdθ the integral becomes (-16/100)∫du/u² and hence the integral becomes (4/25)cscθ + C. (2) Substituting back to x gives (4/25) ((x² +100)^1/2/x) + C as the answer.