Simplifying the Expression: $(x^3\cos(x/2)+1/2)\sqrt{4-x^2}$
In this article, we will simplify the given expression: $(x^3\cos(x/2)+1/2)\sqrt{4-x^2}$. To do this, we will follow the order of operations (PEMDAS) and apply various trigonometric and algebraic identities.
Simplifying the Expression
First, let's analyze the expression: $(x^3\cos(x/2)+1/2)\sqrt{4-x^2}$. We can start by simplifying the square root term.
Simplifying the Square Root Term
The expression inside the square root is $4-x^2$. We can rewrite this as:
$4-x^2 = (2)^2 - (x)^2 = (2+x)(2-x)$
So, the square root term becomes:
$\sqrt{(2+x)(2-x)}$
Now, let's simplify the entire expression:
$(x^3\cos(x/2)+1/2)\sqrt{(2+x)(2-x)}$
Simplifying the Cosine Term
Next, let's focus on the cosine term: $x^3\cos(x/2)$. We can rewrite the cosine term using the trigonometric identity:
$\cos(x/2) = \cos(\frac{x}{2}) = \pm\sqrt{\frac{1+\cos(x)}{2}}$
Since we don't have any information about the quadrant of $x$, we will keep the $\pm$ sign.
Substituting this into the original expression, we get:
$(x^3(\pm\sqrt{\frac{1+\cos(x)}{2}})+1/2)\sqrt{(2+x)(2-x)}$
Combining Like Terms
Now, let's combine the like terms:
$\frac{x^3}{2}(\pm\sqrt{1+\cos(x)})+\frac{1}{2}\sqrt{(2+x)(2-x)}$
Final Simplification
We can't simplify the expression further without additional information about $x$. Therefore, the final simplified expression is:
$\boxed{\frac{x^3}{2}(\pm\sqrt{1+\cos(x)})+\frac{1}{2}\sqrt{(2+x)(2-x)}}$
In conclusion, we have successfully simplified the given expression: $(x^3\cos(x/2)+1/2)\sqrt{4-x^2}$ to its simplest form.