Solve ∫ (from 0 to 2 pi) of t^2d[(1-cost)^2]

$$\int _ { 0 } ^ { 2 \pi } t ^ { 2 } d [ ( 1 - \cos t ) ^ { 2 } ]$$

Evaluate

$$2\pi d\left(t\left(\cos(t)-1\right)\right)^{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int t^{2}d\left(1-\cos(t)\right)^{2}\mathrm{d}x$$

Find the integral of t^{2}d\left(1-\cos(t)\right)^{2} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$t^{2}d\left(1-\cos(t)\right)^{2}x$$

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.

$$2t^{2}d\left(1-\cos(t)\right)^{2}\pi +0t^{2}d\left(1-\cos(t)\right)^{2}$$

Simplify.

$$2d\pi t^{2}\left(\cos(t)-1\right)^{2}$$

Differentiate w.r.t. t

$$4\pi dt\left(\cos(t)-1\right)\left(-t\sin(t)+\cos(t)-1\right)$$