Evaluate : ` int _(-5)^(5) ( x^(17) +sin ^(195)x) dx `

Answer

To evaluate the integral

\[ I = \int_{-5}^{5} \left( x^{17} + \sin^{195}(x) \right) \, dx, \]

we can follow these steps:

### Step 1: Define the function
Let

\[ f(x) = x^{17} + \sin^{195}(x). \]

### Step 2: Check the property of the function
Next, we will check the property of \( f(x) \) by evaluating \( f(-x) \):

\[ f(-x) = (-x)^{17} + \sin^{195}(-x). \]

### Step 3: Simplify \( f(-x) \)
Since \( (-x)^{17} = -x^{17} \) (because 17 is an odd number) and \( \sin(-x) = -\sin(x) \), we have:

\[ \sin^{195}(-x) = (-\sin(x))^{195} = -\sin^{195}(x) \quad \text{(since 195 is also odd)}. \]

Thus,

\[ f(-x) = -x^{17} - \sin^{195}(x) = -\left( x^{17} + \sin^{195}(x) \right) = -f(x). \]

### Step 4: Conclude the symmetry
Since \( f(-x) = -f(x) \), this means that \( f(x) \) is an odd function.

### Step 5: Apply the property of definite integrals
We can use the property of definite integrals for odd functions:

\[ \int_{-a}^{a} f(x) \, dx = 0 \quad \text{if } f(x) \text{ is odd.} \]

### Step 6: Evaluate the integral
Therefore, we conclude that:

\[ I = \int_{-5}^{5} f(x) \, dx = 0. \]

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

\[ \int_{-5}^{5} \left( x^{17} + \sin^{195}(x) \right) \, dx = 0. \]

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