#2**0 **

Using numerical integration techniques,

\(\int _{-2}^2\left(x^3\cos \left(\frac{x}{2}\right)+\frac{1}{2}\right)\sqrt{4-x^2}dx\) would be \(\pi\)

.EpicWater Jan 1, 2020

#5**+2 **

Split the integral into two parts, so you have int(x^{3}cos(x/2)sqrt(4-x^{2}))dx + (1/2)int(sqrt(4-x^{2}))dx

The kernel of the first integral is odd, so when integrating from -2 to +2, the result is zero.

For the second integral make the substitution x = 2theta, say, and it is a simple matter to find the result as pi.

Alan Jan 1, 2020