integrate (x ^ 3 * cos x/2 + 1/2) * sqrt(4 – x ^ 2) dx from – 2 to 2

To integrate the given expression, we can use the method of substitution. Let’s break down the steps:

Step 1: Simplify the expression

The given expression is (x^3 * cos(x/2) + 1/2) * sqrt(4 – x^2).

Step 2: Apply substitution

Let’s substitute u = 4 – x^2. This will help us simplify the expression further.

Differentiating both sides with respect to x, we get du/dx = -2x.

Rearranging, we have dx = -du/(2x).

Step 3: Rewrite the expression in terms of u

Substituting the values of x and dx in terms of u, the expression becomes:

(x^3 * cos(x/2) + 1/2) * sqrt(u) * (-du/(2x))

= (-1/2) * (x^2 * cos(x/2) + 1/2x) * sqrt(u) * du

Step 4: Integrate the expression

Now, we can integrate the expression with respect to u:

??(-1/2) * (x^2 * cos(x/2) + 1/2x) * sqrt(u) * du)

Step 5: Apply limits

To find the definite integral from -2 to 2, we substitute the limits of integration:

??(-1/2) * (x^2 * cos(x/2) + 1/2x) * sqrt(u) * du) from -2 to 2

Step 6: Evaluate the integral

Now, we can evaluate the integral using the antiderivative of the expression. However, finding the antiderivative may not be straightforward, and it might require advanced techniques or numerical methods.

Unfortunately, due to the complexity of the expression, it is not feasible to provide the exact numerical value of the integral without using specialized software or calculators. It is recommended to use numerical integration methods or software to obtain an accurate result.

Please note that the provided solution is a general approach to integrating the given expression. For specific numerical values, it is advisable to use appropriate tools or consult a mathematics expert.

External reference:

– Wolfram Alpha: https://www.wolframalpha.com/