Let the function f and g be integrable on every interval and satisfy the following conditions :
(i) f is odd, g is even function (ii) `g(x)=f(x+5)`
Then which of the following is/are true?

Options

A. `f(x-5)=g(x)`
B. `f(x-5)=-g(x)`
C. `int_(0)^(5)f(t)dt=int_(0)^(5)g(5-t)dt`
D. `int_(0)^(5)f(t)dt=-int_(0)^(5)g(5-t)dt`

Answer

To solve the problem step by step, we will analyze the given conditions and derive the necessary results.

### Step 1: Understanding the properties of the functions
We are given that:
1. \( f(x) \) is an odd function, which means \( f(-x) = -f(x) \).
2. \( g(x) \) is an even function, which means \( g(-x) = g(x) \).
3. The relationship \( g(x) = f(x + 5) \).

### Step 2: Finding \( g(-x) \)
Using the property of \( g(x) \):
\[ g(-x) = f(-x + 5) \]
Since \( f \) is odd, we can rewrite \( f(-x + 5) \):
\[ f(-x + 5) = f(5 - x) = -f(x - 5) \]
Thus, we have:
\[ g(-x) = f(-x + 5) = -f(x - 5) \]

### Step 3: Finding a relationship between \( f(x) \) and \( g(x) \)
Now, substituting \( x = -x \) in the equation \( g(x) = f(x + 5) \):
\[ g(-x) = f(-x + 5) \]
From our earlier results, we know:
\[ g(-x) = -f(x - 5) \]
Setting the two expressions for \( g(-x) \) equal gives us:
\[ f(-x + 5) = -f(x - 5) \]

### Step 4: Finding \( f(x - 5) \)
From the relationship \( g(x) = f(x + 5) \), we can substitute \( x \) with \( x - 5 \):
\[ g(x - 5) = f((x - 5) + 5) = f(x) \]
Thus, we have:
\[ f(x) = g(x - 5) \]

### Step 5: Establishing the relationship between \( f(x) \) and \( g(x) \)
Now, we can derive:
\[ f(x - 5) = g(x) \]
From the odd function property of \( f \):
\[ f(x - 5) = -g(-x) \]

### Step 6: Integration relationships
We can also analyze the integrals:
\[ \int_0^5 f(t) \, dt = \int_0^5 g(5 - t) \, dt \]
This is due to the substitution \( u = 5 - t \).

### Conclusion
From the analysis above, we find:
- \( f(x - 5) = -g(x) \) (which corresponds to option 2).
- \( \int_0^5 f(t) \, dt = \int_0^5 g(5 - t) \, dt \) (which corresponds to option 3).

### Final Answers
- Option 2: True
- Option 3: True