If f(x) is continuous and `int_(0)^(9)f(x)dx=4`, then the value of the integral `int_(0)^(3)x.f(x^(2))dx` is

Options

A. 2
B. 18
C. 16
D. 4

Explanation

To solve the integral \( \int_{0}^{3} x f(x^2) \, dx \) given that \( \int_{0}^{9} f(x) \, dx = 4 \), we can use a substitution method. Here’s the step-by-step solution:

### Step 1: Substitution
Let \( t = x^2 \). Then, we differentiate to find \( dt \):
\[ dt = 2x \, dx \quad \Rightarrow \quad dx = \frac{dt}{2x} \]
From the substitution \( t = x^2 \), we also need to change the limits of integration. When \( x = 0 \), \( t = 0^2 = 0 \) and when \( x = 3 \), \( t = 3^2 = 9 \).

### Step 2: Rewrite the Integral
Now, we can rewrite the integral:
\[ \int_{0}^{3} x f(x^2) \, dx = \int_{0}^{9} x f(t) \cdot \frac{dt}{2x} \]
Notice that \( x \) cancels out:
\[ = \int_{0}^{9} \frac{1}{2} f(t) \, dt \]

### Step 3: Factor Out Constants
We can factor out the constant \( \frac{1}{2} \):
\[ = \frac{1}{2} \int_{0}^{9} f(t) \, dt \]

### Step 4: Substitute the Known Value
We know from the problem statement that:
\[ \int_{0}^{9} f(t) \, dt = 4 \]
Substituting this value into our expression gives:
\[ = \frac{1}{2} \cdot 4 = 2 \]

### Conclusion
Thus, the value of the integral \( \int_{0}^{3} x f(x^2) \, dx \) is \( 2 \).