If `int_(0)^(1) f(x)=M,int_(0)^(1) g(x)dx=N`, then which of the following is correct ?

Options

A. `overset(1)underset(0)int (f(x)+g(x)dx=M+N`
B. `overset(1)underset(0)int (f(x)+g(x)dx=MN`
C. `overset(1)underset(0)int `(1)/(f(x))dx=(1)/(M)`
D. `overset(1)underset(0)int (f(x))/(g(x))dx=(M)/(N)`

Answer

To solve the problem, we need to analyze the given integrals and their properties. We are given:

\[ \int_{0}^{1} f(x) \, dx = M \]
\[ \int_{0}^{1} g(x) \, dx = N \]

We need to determine which of the following statements is correct based on the properties of definite integrals.

### Step 1: Evaluate the sum of the integrals

Using the property of integrals that states:

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

we can write:

\[ \int_{0}^{1} [f(x) + g(x)] \, dx = \int_{0}^{1} f(x) \, dx + \int_{0}^{1} g(x) \, dx \]

Substituting the known values:

\[ \int_{0}^{1} [f(x) + g(x)] \, dx = M + N \]

### Step 2: Evaluate the product of the integrals

For the product of the functions \(f(x)\) and \(g(x)\), we cannot separate the integral in the same way. The integral of a product of two functions is not equal to the product of their integrals:

\[ \int_{0}^{1} f(x) g(x) \, dx \neq \int_{0}^{1} f(x) \, dx \cdot \int_{0}^{1} g(x) \, dx \]

Thus, we cannot conclude anything about \(\int_{0}^{1} f(x) g(x) \, dx\) based solely on \(M\) and \(N\).

### Step 3: Evaluate the integral of the quotient of the functions

Similar to the product, the integral of the quotient of two functions is also not equal to the quotient of their integrals:

\[ \int_{0}^{1} \frac{f(x)}{g(x)} \, dx \neq \frac{\int_{0}^{1} f(x) \, dx}{\int_{0}^{1} g(x) \, dx} \]

This means we cannot derive any useful information about the integral of the quotient from \(M\) and \(N\).

### Conclusion

From the analysis, we conclude that:

1. The integral of the sum of the functions is equal to the sum of their integrals:
\[ \int_{0}^{1} (f(x) + g(x)) \, dx = M + N \]
This is correct.

2. The integral of the product and the integral of the quotient do not yield simple relationships with \(M\) and \(N\).

Thus, the correct statement is that:

\[ \int_{0}^{1} (f(x) + g(x)) \, dx = M + N \]