The value of definite integral: `int_(-2014)^(2014)(dx)/(1+sin^(2015)x+sqrt(1+sin^(4030)x)` equals :

Options

A. 0
B. 2014
C. (2014)^(2)`
D. 4028

Answer

To solve the definite integral

\[ I = \int_{-2014}^{2014} \frac{dx}{1 + \sin^{2015}(x) + \sqrt{1 + \sin^{4030}(x)}} \]

we will follow these steps:

### Step 1: Rationalize the Denominator
We start by rationalizing the denominator:

\[ I = \int_{-2014}^{2014} \frac{1 + \sin^{2015}(x) - \sqrt{1 + \sin^{4030}(x)}}{(1 + \sin^{2015}(x) + \sqrt{1 + \sin^{4030}(x)})(1 + \sin^{2015}(x) - \sqrt{1 + \sin^{4030}(x)}} \, dx \]

This simplifies to:

\[ I = \int_{-2014}^{2014} \frac{(1 + \sin^{2015}(x))^2 - (1 + \sin^{4030}(x))}{(1 + \sin^{2015}(x) + \sqrt{1 + \sin^{4030}(x)})(1 + \sin^{2015}(x) - \sqrt{1 + \sin^{4030}(x)}} \, dx \]

### Step 2: Simplify the Numerator
The numerator simplifies to:

\[ (1 + \sin^{2015}(x))^2 - (1 + \sin^{4030}(x)) = 1 + 2\sin^{2015}(x) + \sin^{4030}(x) - 1 - \sin^{4030}(x) = 2\sin^{2015}(x) \]

Thus, we have:

\[ I = \int_{-2014}^{2014} \frac{2\sin^{2015}(x)}{(1 + \sin^{2015}(x) + \sqrt{1 + \sin^{4030}(x)})} \, dx \]

### Step 3: Analyze the Function
Notice that \(\sin(-x) = -\sin(x)\), which implies that \(\sin^{2015}(-x) = -\sin^{2015}(x)\). Therefore, the function \(f(x) = \frac{\sin^{2015}(x)}{(1 + \sin^{2015}(x) + \sqrt{1 + \sin^{4030}(x)})}\) is an odd function.

### Step 4: Use the Property of Odd Functions
Since \(f(x)\) is odd, we have:

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

for any \(a\). Thus,

\[ I = \int_{-2014}^{2014} f(x) \, dx = 0 \]

### Step 5: Conclusion
The value of the definite integral is:

\[ \boxed{0} \]