Evaluate the following integrals:
`int_(-1)^(1)x^(99)cos^(4)x dx`

Answer

To evaluate the integral \( \int_{-1}^{1} x^{99} \cos^{4} x \, dx \), we can follow these steps:

### Step 1: Identify the function and check its properties
Let \( f(x) = x^{99} \cos^{4} x \).

### Step 2: Check if \( f(x) \) is an odd or even function
To determine whether \( f(x) \) is odd or even, we need to evaluate \( f(-x) \):
\[ f(-x) = (-x)^{99} \cos^{4}(-x) \]
Since \( (-x)^{99} = -x^{99} \) (because 99 is odd) and \( \cos(-x) = \cos(x) \) (because cosine is an even function), we have:
\[ f(-x) = -x^{99} \cos^{4} x \]

### Step 3: Determine the nature of \( f(x) \)
From the above calculation, we see that:
\[ f(-x) = -f(x) \]
This means that \( f(x) \) is an odd function.

### Step 4: Use the property of integrals of odd functions
The property of integrals states that the integral of an odd function over a symmetric interval around zero is zero:
\[ \int_{-a}^{a} f(x) \, dx = 0 \quad \text{if } f(x) \text{ is odd} \]
In our case, since \( f(x) \) is odd and we are integrating from \(-1\) to \(1\), we can conclude:
\[ \int_{-1}^{1} x^{99} \cos^{4} x \, dx = 0 \]

### Final Answer
Thus, the value of the integral is:
\[ \int_{-1}^{1} x^{99} \cos^{4} x \, dx = 0 \]

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