Evaluate
1-\cos(1)\approx 0.459697694
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\int \sin(t)\mathrm{d}t
Evaluate the indefinite integral first.
-\cos(t)
Use \int \sin(t)\mathrm{d}t=-\cos(t) from the table of common integrals to obtain the result.
-\cos(1)+\cos(0)
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.
-\cos(1)+1
Simplify.
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