What is the integral of f(x)=sin(x/2)/x?

Question

What is the integral of f(x)=sin(x/2)/x?

Answer 1

As an indefinite integral, sin(x)/x is non-integrable. You can expand sin(x/2) using its Taylor series, and integrate that, but you're then dealing with an infinite series. If you're looking for a definite integral, you can use the Taylor series to approximate fairly accurately.

 

The value of ∫₀^∞sin(x)/x dx is known to be π/2, so if you have those limits, you can use a u-substitution with u=x/2 and the known integral to find the value.