Calculus 2 Evaluate the integral (7.3 #2)

Question

Evaluate the integral using the indicated trigonometric substitution. (Use C for the constant of integration.)

x2-16/x dx, x=4sec(theta)
It is a fraction the numerator is square root x2-16, and the denominator is x. Please show detailed work and box the final answer, thank you.

Answer 1

We start with the indicated substitution and find dx. We have:

x = 4sec(z)

z = sec^-1(x/4)

dx = 4sec(z)tan(z)dz

Then, we have:

∫sqrt(x² - 16)dx/x = 16 ∫sqrt(sec²z - 1)sec(z)tan(z)dz/[4sec(z)] = 4 ∫sqrt(sec²z - 1)tan(z)dz = 4 ∫tan²z dz

Now, we have:

4 ∫ tan²z dz = 4 ∫ (sec²z - 1)dz = 4(tan z - z) + C = 4tan(sec^-1(x/4)) - 4sec^-1(x/4) + C

Simplify the nested trig function by setting up a triangle. The hypotenuse is x and the adjacent side is 4, so the opposite side is sqrt(x² - 16). Therefore, our final answer is:

∫sqrt(x² - 16)dx/x = = 4sqrt(x² - 16)/x - 4 sec^-1(x/4) + C