Hit "Shift Return" whith the cursor in each line, to make it evaluate the line. (Or use "Evaluate Notebook" in the Evaluation menu to do them all at once) 
The examples below contain many of the main features of Mathematica that wil be useful in solving physics problems.. 
Syntax matters - watch for square brackets vs curly brackets vs parentheses, capitals, colons, etc. 
All Mathematica functions start with a capital letters. Function arguments are in square brackets, separated by commas. If you define your functions beginning with a lowercase letter, you will not see any conflicts.

Let' s say you want to compute an integral of the sine function, Sin (n π y) where y goes from zero to one. This is done on the line below. Move the cursor to that line and hit "Shift-Return" or "Shift-Enter"

2*Integrate[Sin[n* Pi* y],{y,0,1}]

Here is a slightly more elaborate Integral which doesn' t quite do what we want.

Integrate[x*Sin[n*Pi*x/a]^2 / a,{x, 0, a}]

The issue here is that Mathematica does not assume that n is an integer. In the example below, we tell Mathematica to assume that n is an integer.

Integrate[x*Sin[n*Pi*x/a]^2 / a,{x, 0, a},Assumptions->n∈Integers]

Calculate an aray of coefficients. Here f is actually a function which when called will evaluate it's argument(s) and provide its value as an answer. Notice the underbar on the left side but not on the right side.

f[x_]:= Sin[2*x]

f[Pi/8]

If your function can only take a particular type of input, say an integer, you can tell Mathematica to only evaluate the function when given proper inputs.

g[x_,n_Integer]:=Sin[n*x]

Try evaluating the two statements below

g[Pi/2,3]

g[Pi,3/2]

Finally, you can integrate functions and integrate in functions:

Integrate[g[x,4], {x, 0, 1}]

N[%]

h[a_]:=Integrate[x*Sin[n*Pi*x/a]^2 / a, {x, 0, a}, 
 Assumptions -> n ∈ Integers]

h[1]