Integration of Seemingly Simple Function

Question

Task:
Find a function $f$ such that $f(2) = -1$ and $f'(x) = \frac {sin(x)} {x}$

$$\int \frac {sin(x)} {x} dx = ? + C$$

Research:
After searching online for the value of the indefinite integral $\int \frac {sin(x)} {x} dx$, I have discovered it involves the power series (which we have not yet learned).
Therefore, I believe there is another way to solve the problem above.

Any hints to solving the integral (without the power series) or another method to solve the problem is appreciated.

Answer 1

The "correct" answer to this ubiquitous question in a typical calculus course is $$f(x) = \int_2^x \frac{\sin(t)}{t}dt - 1.$$ It's designed to make sure you understand the fundamental theorem of calculus and integration limits, not to test your integration skills.

(As others have commented, $\sin(x)/x$ has no elementary antiderivative.)

Answer 2

The sine integral function is defined as

$$\mathrm {Si}(z)=\int^z_0 \frac {\sin x}{x}dx$$