`f(x)` is a continuous function for all real values of `x` and satisfies `int_n^(n+1)f(x)dx=(n^2)/2AAn in Idot` Then `int_(-3)^5f(|x|)dx` is equal to `(19)/2` (b) `(35)/2` (c) `(17)/2` (d) none of these

Options

A. `19/2`
B. `35/2`
C. `17/2`
D. `37/2`

Answer

To solve the problem, we need to evaluate the integral \( \int_{-3}^{5} f(|x|) \, dx \) given that \( \int_{n}^{n+1} f(x) \, dx = \frac{n^2}{2} \) for integer values of \( n \).

### Step-by-Step Solution

1. **Break Down the Integral**:
We can express the integral \( \int_{-3}^{5} f(|x|) \, dx \) as:
\[ \int_{-3}^{5} f(|x|) \, dx = \int_{-3}^{0} f(-x) \, dx + \int_{0}^{5} f(x) \, dx \]

2. **Change of Variables for the First Integral**:
For the first integral \( \int_{-3}^{0} f(-x) \, dx \), we can use the substitution \( t = -x \). Thus, when \( x = -3 \), \( t = 3 \) and when \( x = 0 \), \( t = 0 \). The differential changes as \( dx = -dt \):
\[ \int_{-3}^{0} f(-x) \, dx = \int_{3}^{0} f(t) (-dt) = \int_{0}^{3} f(t) \, dt \]

3. **Combine the Integrals**:
Now we can rewrite our original integral:
\[ \int_{-3}^{5} f(|x|) \, dx = \int_{0}^{3} f(x) \, dx + \int_{0}^{5} f(x) \, dx \]
This can be simplified to:
\[ = \int_{0}^{3} f(x) \, dx + \int_{3}^{5} f(x) \, dx \]

4. **Split the Integral**:
We can further split the integral:
\[ = \int_{0}^{3} f(x) \, dx + \int_{3}^{4} f(x) \, dx + \int_{4}^{5} f(x) \, dx \]

5. **Apply the Given Property**:
We know from the problem statement that:
\[ \int_{n}^{n+1} f(x) \, dx = \frac{n^2}{2} \]
Therefore, we can calculate:
- For \( n = 0 \):
\[ \int_{0}^{1} f(x) \, dx = \frac{0^2}{2} = 0 \]
- For \( n = 1 \):
\[ \int_{1}^{2} f(x) \, dx = \frac{1^2}{2} = \frac{1}{2} \]
- For \( n = 2 \):
\[ \int_{2}^{3} f(x) \, dx = \frac{2^2}{2} = 2 \]
- For \( n = 3 \):
\[ \int_{3}^{4} f(x) \, dx = \frac{3^2}{2} = \frac{9}{2} \]
- For \( n = 4 \):
\[ \int_{4}^{5} f(x) \, dx = \frac{4^2}{2} = 8 \]

6. **Sum the Integrals**:
Now we can sum these results:
\[ \int_{0}^{3} f(x) \, dx = 0 + \frac{1}{2} + 2 = \frac{5}{2} \]
\[ \int_{3}^{5} f(x) \, dx = \frac{9}{2} + 8 = \frac{9}{2} + \frac{16}{2} = \frac{25}{2} \]

7. **Final Calculation**:
Therefore, the total integral is:
\[ \int_{-3}^{5} f(|x|) \, dx = \frac{5}{2} + \frac{25}{2} = \frac{30}{2} = 15 \]

### Conclusion
The value of \( \int_{-3}^{5} f(|x|) \, dx \) is \( 15 \).