trig substitution for an interval

Question

evaluate the integral: 26(sqrt(x^2 - 25)) / (x^4) dx

the trig substitution for this is: 5sec(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 answer these, i am struggling. it would help if you could also number each of your answers with #1, #2, and #3.

Answer 1

1) Let x = 5secθ Integrand become 26*sqrt(25sec²θ-25)/5⁴sec⁴θ dx = 5secθtanθdθ and the sqrt term becomes 5tanθ.

New integrand is 26/25 * tan²θ/sec³θ dθ which does not look so great. But if you put it in terms of sin and cos...

26/25 * sin²θcosθ dθ which is an easy integral because its a udu situation (cos is derivative of sin)

2) Do above integral

3) Since you know sec θ = x/5, you can make the right triangle with hypotenuse x/5 and horizontal side 1 in order to find sinθ = sqrt((x/5)²-1)/(x/5) and sub into (2)

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