Statement-1: `int_(0)^(pi//2) (1)/(1+tan^(3)x)dx=(pi)/(4)`
Statement-2: `int_(0)^(a) f(x)dx=int_(0)^(a) f(a+x)dx`

Options

A. Statement-1 is true, Statement-2 is True,Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Explanation

To solve the given problem, we will analyze both statements step by step.

### Step 1: Analyze Statement 1
We need to evaluate the integral:

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{1}{1 + \tan^3 x} \, dx \]

### Step 2: Use the property of definite integrals
We will use the property of definite integrals that states:

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

In this case, we set \( a = \frac{\pi}{2} \):

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{1}{1 + \tan^3\left(\frac{\pi}{2} - x\right)} \, dx \]

### Step 3: Simplify the integral
Using the identity \( \tan\left(\frac{\pi}{2} - x\right) = \cot x \), we can rewrite the integral:

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{1}{1 + \cot^3 x} \, dx \]

### Step 4: Rewrite the cotangent
Recall that \( \cot x = \frac{1}{\tan x} \). Thus, we have:

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\tan^3 x}{\tan^3 x + 1} \, dx \]

### Step 5: Add the two expressions for I
Now we have two expressions for \( I \):

1. \( I = \int_{0}^{\frac{\pi}{2}} \frac{1}{1 + \tan^3 x} \, dx \)
2. \( I = \int_{0}^{\frac{\pi}{2}} \frac{\tan^3 x}{\tan^3 x + 1} \, dx \)

Adding these two equations:

\[ 2I = \int_{0}^{\frac{\pi}{2}} \left( \frac{1}{1 + \tan^3 x} + \frac{\tan^3 x}{\tan^3 x + 1} \right) \, dx \]

### Step 6: Simplify the integrand
The integrand simplifies to:

\[ \frac{1 + \tan^3 x}{1 + \tan^3 x} = 1 \]

Thus, we have:

\[ 2I = \int_{0}^{\frac{\pi}{2}} 1 \, dx = \frac{\pi}{2} \]

### Step 7: Solve for I
Dividing both sides by 2 gives:

\[ I = \frac{\pi}{4} \]

### Conclusion for Statement 1
Therefore, Statement 1 is true:

\[ \int_{0}^{\frac{\pi}{2}} \frac{1}{1 + \tan^3 x} \, dx = \frac{\pi}{4} \]

### Step 8: Analyze Statement 2
Statement 2 states:

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

This statement is generally not true. The correct property is:

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

### Conclusion for Statement 2
Thus, Statement 2 is false.

### Final Conclusion
- Statement 1 is true.
- Statement 2 is false.