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Differentiate w.r.t. t
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\int \left(\cos(t)+t\sin(t)\right)d\left(\sin(t)-t\cos(t)\right)\mathrm{d}x
Evaluate the indefinite integral first.
\left(\cos(t)+t\sin(t)\right)d\left(\sin(t)-t\cos(t)\right)x
Find the integral of \left(\cos(t)+t\sin(t)\right)d\left(\sin(t)-t\cos(t)\right) using the table of common integrals rule \int a\mathrm{d}x=ax.
2\left(\cos(t)+t\sin(t)\right)d\left(\sin(t)-t\cos(t)\right)\pi +0\left(\cos(t)+t\sin(t)\right)d\left(\sin(t)-t\cos(t)\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\left(\cos(t)+t\sin(t)\right)d\left(\sin(t)-t\cos(t)\right)\pi
Simplify.