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 problem, we will analyze both statements step by step.

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

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

**Step 1: Use the property of definite integrals**

We can use the property of definite integrals that states:

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

In our case, we set \( a = \frac{\pi}{2} \), so we rewrite the integral as:

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

**Step 2: Simplify the integrand**

Using the identity \( \tan\left(\frac{\pi}{2} - x\right) = \cot x \), we have:

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

Now, we can rewrite \( \cot^3 x \) as \( \frac{1}{\tan^3 x} \):

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

**Step 3: Add the two integrals**

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 both expressions:

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

**Step 4: Simplify the integrand**

The sum simplifies to:

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

Thus, we have:

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

**Step 5: Solve for \( I \)**

Now, dividing both sides by 2:

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

Thus, Statement 1 is correct:

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

### Statement 2:
We need to evaluate the statement:

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

**Step 1: Change of variable**

Let \( t = a + x \). Then, \( dt = dx \).

**Step 2: Change the limits**

When \( x = 0 \), \( t = a \) and when \( x = a \), \( t = 2a \). Thus, we rewrite the integral:

\[ \int_{0}^{a} f(a + x) \, dx = \int_{a}^{2a} f(t) \, dt \]

**Step 3: Compare the integrals**

The left-hand side is:

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

The right-hand side is:

\[ \int_{a}^{2a} f(t) \, dt \]

These two integrals are not necessarily equal unless \( f(x) \) has specific properties. Therefore, Statement 2 is incorrect.

### Conclusion:
- Statement 1 is true: \( \int_{0}^{\frac{\pi}{2}} \frac{1}{1 + \tan^3 x} \, dx = \frac{\pi}{4} \)
- Statement 2 is false: \( \int_{0}^{a} f(x) \, dx \neq \int_{0}^{a} f(a + x) \, dx \)