Statement -1 : If `Delta(x)=|{:(f_1(x),f_2(x),f_3(x)),(a_2,b_2,c_2),(a_3,b_3,c_3):}|` where, `f_1 , f_2,f_3` are different functions and `a_2,b_2,c_2 and a_3,b_3,c_3` are constants then `intDelta(x)dx=|{:(intf_1(x)dx,intf_2(x)dx,intf_3(x)dx),(a_2,b_2,c_2),(a_3,b_3,c_3):}|`
because
Statement-2 : Integration of sum of several functions is equal to the sum of integration of individual function.

Options

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

Answer

To solve the problem, we need to analyze the statements provided and verify their correctness step by step.

### Step 1: Understand the Determinant
We have a determinant defined as:
\[ \Delta(x) = \begin{vmatrix} f_1(x) & f_2(x) & f_3(x) \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} \]
where \(f_1, f_2, f_3\) are different functions of \(x\) and \(a_2, b_2, c_2, a_3, b_3, c_3\) are constants.

### Step 2: Calculate the Determinant
Using the properties of determinants, we can expand \(\Delta(x)\):
\[ \Delta(x) = f_1(x) \begin{vmatrix} b_2 & c_2 \\ b_3 & c_3 \end{vmatrix} - f_2(x) \begin{vmatrix} a_2 & c_2 \\ a_3 & c_3 \end{vmatrix} + f_3(x) \begin{vmatrix} a_2 & b_2 \\ a_3 & b_3 \end{vmatrix} \]
Calculating the 2x2 determinants:
\[ \Delta(x) = f_1(x)(b_2c_3 - c_2b_3) - f_2(x)(a_2c_3 - a_3c_2) + f_3(x)(a_2b_3 - a_3b_2) \]

### Step 3: Integrate the Determinant
Now we need to integrate \(\Delta(x)\):
\[ \int \Delta(x) \, dx = \int \left[ f_1(x)(b_2c_3 - c_2b_3) - f_2(x)(a_2c_3 - a_3c_2) + f_3(x)(a_2b_3 - a_3b_2) \right] dx \]
Using the linearity of integration, we can separate the integrals:
\[ \int \Delta(x) \, dx = (b_2c_3 - c_2b_3) \int f_1(x) \, dx - (a_2c_3 - a_3c_2) \int f_2(x) \, dx + (a_2b_3 - a_3b_2) \int f_3(x) \, dx \]

### Step 4: Formulate the Result
Thus, we can express the integral of the determinant as:
\[ \int \Delta(x) \, dx = \begin{vmatrix} \int f_1(x) \, dx & \int f_2(x) \, dx & \int f_3(x) \, dx \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} \]

### Conclusion
From the above steps, we can conclude that Statement 1 is indeed correct as it aligns with the properties of determinants and integration.