Which of the following is incorrect ?

Options

A. (a) `int_(a+c)^(b+c)f(x)dx=int_(a)^(b)f(x+c)dx`
B. (b) `int_(ac)^(bc)f(x)dx=cint_(a)^(b)f(cx)dx`
C. (c) `int_(-a)^(a)f(x)dx=1/2int_(-a)^(a)f(x)+f(-x)dx`
D. (d) none of these

Explanation

To determine which of the given options is incorrect, we will analyze each option step by step.

### Step 1: Analyze Option A
**LHS:** \(\int_{a+c}^{b+c} f(x) \, dx\)

1. Substitute \(x = t + c\):
- When \(x = a + c\), \(t = a\)
- When \(x = b + c\), \(t = b\)

2. Change the integral:
\[ \int_{a}^{b} f(t + c) \, dt \]

3. Since \(dx = dt\), we have:
\[ \int_{a}^{b} f(t + c) \, dt = \int_{a}^{b} f(x + c) \, dx \]

**RHS:** \(\int_{a}^{b} f(x + c) \, dx\)

Thus, LHS = RHS, so **Option A is correct.**

### Step 2: Analyze Option B
**LHS:** \(\int_{ac}^{bc} f(x) \, dx\)

1. Substitute \(x = tc\):
- When \(x = ac\), \(t = a\)
- When \(x = bc\), \(t = b\)

2. Change the integral:
\[ \int_{a}^{b} f(tc) \, c \, dt \]

3. Factor out \(c\):
\[ c \int_{a}^{b} f(tc) \, dt \]

**RHS:** \(c \int_{a}^{b} f(xc) \, dx\)

Thus, LHS = RHS, so **Option B is correct.**

### Step 3: Analyze Option C
**LHS:** \(\int_{-a}^{a} f(x) \, dx\)

1. Let \(I = \int_{-a}^{a} f(x) \, dx\).

2. Substitute \(x = -t\):
- When \(x = -a\), \(t = a\)
- When \(x = a\), \(t = -a\)

3. Change the integral:
\[ I = \int_{a}^{-a} f(-t)(-dt) = \int_{-a}^{a} f(-t) \, dt \]

4. Thus, we have:
\[ I = \int_{-a}^{a} f(-x) \, dx \]

5. Adding both integrals:
\[ 2I = \int_{-a}^{a} (f(x) + f(-x)) \, dx \]
\[ I = \frac{1}{2} \int_{-a}^{a} (f(x) + f(-x)) \, dx \]

**RHS:** \(\frac{1}{2} \int_{-a}^{a} (f(x) + f(-x)) \, dx\)

Thus, LHS = RHS, so **Option C is correct.**

### Step 4: Analyze Option D
Since options A, B, and C are all correct, we conclude that **Option D must be the incorrect one**.

### Final Conclusion
The incorrect option is **Option D**.