Solve ∫ (from 1 to frac { 15) of {2}}text{arrsint}dt

$$\int _ { 1 } ^ { \frac { 15 } { 2 } } \text { arrsint } d t$$

Evaluate

$$\left(\cos(1)-\cos(\frac{15}{2})\right)ar^{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int a\sin(t)r^{2}\mathrm{d}t$$

Factor out the constant using \int af\left(t\right)\mathrm{d}t=a\int f\left(t\right)\mathrm{d}t.

$$ar^{2}\int \sin(t)\mathrm{d}t$$

Use \int \sin(a)\mathrm{d}a=-\cos(a) from the table of common integrals to obtain the result.

$$-ar^{2}\cos(t)$$

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.

$$-ar^{2}\cos(\frac{15}{2})+ar^{2}\cos(1)$$

Simplify.

$$a\left(-\cos(\frac{15}{2})+\cos(1)\right)r^{2}$$

Differentiate w.r.t. a

$$\left(\cos(1)-\cos(\frac{15}{2})\right)r^{2}$$