Evaluate the Integral using the indicated trigonometric substitution. (Use C for the Constant of Integration)

Question

Evaluate the Integral using the indicated trigonometric substitution. (Use C for the Constant of Integration). Please show all steps. If possible please attach a video going through the problem. I found the solution but my answer is different than the correct answer so I would like to see someone go through the problem to see where my mistake is.

Answer 1

First set up the problem:

Then sub x = (5/2)sec(θ) or sec(θ) = (2/5)x.

Then dx = (5/2)tan(θ)sec(θ) dθ

The integral becomes:

Simplifying:

5∫tan(θ)√(sec²(θ) - 1) dθ

Then applying the Pythagorean Identity tan2(θ) = sec2(θ) - 1 and taking the square root we get:

5∫tan²(θ) dθ

Again, sub sec²(θ) - 1 for tan²(θ) to get:

5∫sec²(θ) - 1 dθ

Take the antiderivative to get:

5tan(θ) - 5θ + C

Using our original trig sub: sec(θ) = (2/5)x re-write as sec(θ) = (2x)/5 and consider the associated triangle where hypotenuse = 2x and adjacent side = 5. That makes the opposite side = √(4x² - 25).

Using these relationships, tan(θ) = opp/adj = √(4x² - 25)/5 and it also makes θ = sec^-1[(2x)/5]

Sub these in to get the answer you noted.