integrate (x^3 cos (x/2) + 1/2) (4-x^2)^1/2 dx

To solve the integral \(\int \left( x^3 \cos \left( \frac{x}{2} \right) + \frac{1}{2} \right) \sqrt{4 - x^2} \, dx\), we will proceed step by step.

### Step 1: Split the Integral
First, break the integral into two parts:

\[
I = \int \left( x^3 \cos \left( \frac{x}{2} \right) + \frac{1}{2} \right) \sqrt{4 - x^2} \, dx
\]

This can be written as:

\[
I = \int x^3 \cos \left( \frac{x}{2} \right) \sqrt{4 - x^2} \, dx + \int \frac{1}{2} \sqrt{4 - x^2} \, dx
\]

We now handle each term separately.

### Step 2: Solve the Second Integral
Start with the simpler term:

\[
\int \frac{1}{2} \sqrt{4 - x^2} \, dx
\]

This is a standard integral, and it can be solved by a trigonometric substitution:

Let \( x = 2 \sin \theta \), so that \( dx = 2 \cos \theta \, d\theta \), and \( \sqrt{4 - x^2} = 2 \cos \theta \). The integral becomes:

\[
\frac{1}{2} \int 2 \cos^2 \theta \cdot 2 \cos \theta \, d\theta = 2 \int \cos^2 \theta \, d\theta
\]

Now use the identity \( \cos^2 \theta = \frac{1 + \cos(2\theta)}{2} \):

\[
2 \int \cos^2 \theta \, d\theta = \int (1 + \cos(2\theta)) \, d\theta
\]

This integrates to:

\[
\theta + \frac{1}{2} \sin(2\theta) + C
\]

Converting back to \(x\), recall that \( \theta = \arcsin \left( \frac{x}{2} \right) \), and we get:

\[
\frac{x}{2} \sqrt{4 - x^2} + C_1
\]

### Step 3: First Integral
Now handle the more complicated term:

\[
\int x^3 \cos \left( \frac{x}{2} \right) \sqrt{4 - x^2} \, dx
\]

This integral is more challenging and may require advanced techniques such as integration by parts or a series expansion. For now, this can be computed using numerical methods or specialized software for exact solutions, as it does not have a simple closed form.

### Step 4: Combine the Results
Thus, the solution to the integral is:

\[
I = \frac{x}{2} \sqrt{4 - x^2} + \text{(solution to the first integral)} + C
\]

Let me know if you'd like help finding the numerical value of the first term or clarifying any step.

---

Would you like more details or have any questions?  
Here are some related problems to explore:

1. Integrate \( \int \sqrt{4 - x^2} \, dx \).
2. Solve \( \int x^2 \cos(x) \, dx \).
3. Compute \( \int \sin(x) \sqrt{1 - x^2} \, dx \).
4. Find the integral \( \int x^3 e^x \, dx \).
5. Evaluate \( \int x^3 \sqrt{4 - x^2} \, dx \).

**Tip:** Breaking a complex integral into smaller parts is often key to finding a solution.

Solve the complex integral ∫(x^3 cos(x/2) + 1/2) sqrt(4 - x^2) dx. This problem involves advanced calculus techniques, including trigonometric substitution and integration by parts. Suitable for university-level students studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation