Suppose $\int f(x) d x=\frac{16}{x}+12 \cdot 2^{-x}+16 \tan \left(\frac{\pi}{4 x}\right)+7 \cos \left(\frac{\pi}{2 x}\right)$
Determine $\int_{1}^{\infty} f(x) d x$.
The integral converges to

Step 1 ：Suppose the integral of function f(x) is given by \(\int f(x) d x=\frac{16}{x}+12 \cdot 2^{-x}+16 \tan \left(\frac{\pi}{4 x}\right)+7 \cos \left(\frac{\pi}{2 x}\right)\)

Step 2 ：We want to determine the value of \(\int_{1}^{\infty} f(x) d x\)

Step 3 ：The integral of a function from 1 to infinity is the limit as b goes to infinity of the integral from 1 to b

Step 4 ：We can find the antiderivative of f(x) and evaluate it at 1 and b, then take the limit as b goes to infinity

Step 5 ：The antiderivative of f(x) is given in the question

Step 6 ：Let's denote the antiderivative as F, so F = \(7\cos(\frac{\pi}{2x}) + 16\tan(\frac{\pi}{4x}) + \frac{16}{x} + \frac{12}{2^x}\)

Step 7 ：Taking the limit as b goes to infinity, we find that the value is -31

Step 8 ：So, the integral converges to \(\boxed{-31}\)