Question

Find the indefinite integral by using the substitution x = 4 sec(θ). (Use C for the constant of integration.) x2 − 16 x d

Answer 1

Answer:

$\frac{x^2-16}{2} + 16ln\frac{4}{x} +16C$

Step-by-step explanation:

Given the indefinite integral $\int\limits{\frac{x^2-16}{x} } \, dx$, using the substitute

x = 4 sec(θ)...1

The integral can be calculated as thus;

First let us diffrentiate the substitute function with respect to θ

dx/dθ = 4secθtanθ

dx =  4secθtanθdθ... 2

Substituting equation 1 and 2 into the integral function we will have;

$\int\limits{\frac{(4sec \theta)^2-16}{4sec \theta} } \, 4sec \theta tan \theta d \theta\\\int\limits{\frac{16sec^2 \theta-16}{4sec \theta} } \, 4sec \theta tan \theta d \theta\\\int\limits{\frac{(16(sec^2 \theta-1)}{4sec \theta} } \, 4sec \theta tan \theta d \theta\\\\from \ trig \ identity,\ sec^2 \theta - 1 = tan^\theta\\\\\int\limits{\frac{16 tan^2 \theta}{4sec \theta} } \, 4sec \theta tan \theta d \theta\\\\\int\limits 16 tan^3 \theta d \theta$

Find the remaining solution in the attachment.