Solve ∫ (from 0 to 2 pi) of (cost+tsint)d(sint-tcost)

$$\int _ { 0 } ^ { 2 \pi } ( \cos t + t \sin t ) d ( \sin t - t \cos t )$$

Evaluate

$$2\pi d\left(-t\cos(t)+\sin(t)\right)\left(t\sin(t)+\cos(t)\right)$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \left(\cos(t)+t\sin(t)\right)d\left(\sin(t)-t\cos(t)\right)\mathrm{d}x$$

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.

$$\left(\cos(t)+t\sin(t)\right)d\left(\sin(t)-t\cos(t)\right)x$$

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.

$$2\left(\cos(t)+t\sin(t)\right)d\left(\sin(t)-t\cos(t)\right)\pi +0\left(\cos(t)+t\sin(t)\right)d\left(\sin(t)-t\cos(t)\right)$$

Simplify.

$$2\left(\cos(t)+t\sin(t)\right)d\left(\sin(t)-t\cos(t)\right)\pi$$

Differentiate w.r.t. t

$$2\pi dt\left(\sin(t)\left(t\sin(t)+\cos(t)\right)+\cos(t)\left(-t\cos(t)+\sin(t)\right)\right)$$