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The range of x values where you want the plot should look like (x, 0, 10) instead of x, 0, 10.

The following works for me:

sage: a = 0.5
sage: x, t = SR.var('x t')
sage: plot(integrate(cos(x*t), t, 0, sin(a)), (x, 0, 10))

Or you can use a Python function rather than a Sage symbolic function, and describe the range as (0, 10).

sage: plot(lambda x: integrate(cos(x*t), t, 0, sin(a)), (0, 10))

The syntax lambda x: expression involving x is the Python equivalent of the mathematical notation $x \mapsto \text{expression involving } x$.