(x^3*cos(x/2)+1/2)*sqrt(4-x^2) Answer

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Jul 03, 2024

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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.