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\int \sin(x)-1\mathrm{d}x
Evaluate the indefinite integral first.
\int \sin(x)\mathrm{d}x+\int -1\mathrm{d}x
Integrate the sum term by term.
-\cos(x)+\int -1\mathrm{d}x
Use \int \sin(x)\mathrm{d}x=-\cos(x) from the table of common integrals to obtain the result.
-\cos(x)-x
Find the integral of -1 using the table of common integrals rule \int a\mathrm{d}x=ax.
-\cos(2\pi )-2\pi -\left(-\cos(\pi )-\pi \right)
The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.
-2-\pi
Simplify.