`int_(-3pi//2)^(-pi//2){(x+pi)^(3)+cos^(2)(x+3pi)}dx`, is

Options

A. `(pi^(4))/(32)`
B. `(pi^(4))/(32)+(pi)/(2)`
C. `(pi)/(2)`
D. `(pi)/(4)-1`

Explanation

To solve the integral

\[ \int_{-\frac{3\pi}{2}}^{-\frac{\pi}{2}} \left( (x + \pi)^3 + \cos^2(x + 3\pi) \right) dx, \]

we will follow these steps:

### Step 1: Substitution
Let \( t = x + \pi \). Then, \( dx = dt \). We also need to change the limits of integration:
- When \( x = -\frac{3\pi}{2} \), \( t = -\frac{3\pi}{2} + \pi = -\frac{\pi}{2} \).
- When \( x = -\frac{\pi}{2} \), \( t = -\frac{\pi}{2} + \pi = \frac{\pi}{2} \).

Thus, the integral becomes:

\[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( t^3 + \cos^2(t + 2\pi) \right) dt. \]

### Step 2: Simplifying the Cosine Term
Using the periodicity of cosine, we have:

\[ \cos(t + 2\pi) = \cos(t). \]

So, we can rewrite the integral as:

\[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( t^3 + \cos^2(t) \right) dt. \]

### Step 3: Split the Integral
We can split the integral into two parts:

\[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} t^3 dt + \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos^2(t) dt. \]

### Step 4: Evaluate the First Integral
The first integral, \( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} t^3 dt \), is an odd function over a symmetric interval about zero. Therefore, it evaluates to zero:

\[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} t^3 dt = 0. \]

### Step 5: Evaluate the Second Integral
For the second integral, we can use the identity \( \cos^2(t) = \frac{1 + \cos(2t)}{2} \):

\[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos^2(t) dt = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{1 + \cos(2t)}{2} dt = \frac{1}{2} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} dt + \frac{1}{2} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos(2t) dt. \]

The first integral evaluates to:

\[ \frac{1}{2} \left[ t \right]_{-\frac{\pi}{2}}^{\frac{\pi}{2}} = \frac{1}{2} \left( \frac{\pi}{2} - \left(-\frac{\pi}{2}\right) \right) = \frac{1}{2} \cdot \pi = \frac{\pi}{2}. \]

The second integral, \( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos(2t) dt \), also evaluates to zero because \( \cos(2t) \) is an even function over a symmetric interval about zero.

### Step 6: Combine Results
Combining the results from both integrals, we have:

\[ 0 + \frac{\pi}{2} = \frac{\pi}{2}. \]

### Final Answer
Thus, the value of the integral is:

\[ \int_{-\frac{3\pi}{2}}^{-\frac{\pi}{2}} \left( (x + \pi)^3 + \cos^2(x + 3\pi) \right) dx = \frac{\pi}{2}. \]