`(1)/(c )int_(ac)^(bc)f((x)/(c ))dx=`

Options

A. `c int_(a)^(b)f(cx)dx`
B. `(1)/(c )int_(a)^(b)f((x)/(c ))dx`
C. `(1)/(c )int_(a)^(b)f(cx)dx`
D. `c int_(a)^(b)f((x)/(c ))dx`

Answer

To solve the integral

\[ \frac{1}{c} \int_{ac}^{bc} f\left(\frac{x}{c}\right) \, dx, \]

we will perform a substitution to simplify the expression.

### Step 1: Substitute \( t = \frac{x}{c} \)

Let \( t = \frac{x}{c} \). Then, we can express \( x \) in terms of \( t \):

\[ x = ct. \]

### Step 2: Differentiate to find \( dx \)

Differentiating both sides with respect to \( t \):

\[ dx = c \, dt. \]

### Step 3: Change the limits of integration

Now, we need to change the limits of integration. When \( x = ac \):

\[ t = \frac{ac}{c} = a. \]

When \( x = bc \):

\[ t = \frac{bc}{c} = b. \]

### Step 4: Substitute into the integral

Now substitute \( x \) and \( dx \) in the integral:

\[ \frac{1}{c} \int_{ac}^{bc} f\left(\frac{x}{c}\right) \, dx = \frac{1}{c} \int_{a}^{b} f(t) \cdot (c \, dt). \]

### Step 5: Simplify the expression

The \( c \) in the numerator and denominator cancels out:

\[ = \int_{a}^{b} f(t) \, dt. \]

### Step 6: Change \( t \) back to \( x \)

Since \( t \) was just a substitution, we can rename it back to \( x \):

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

### Final Result

Thus, we conclude that:

\[ \frac{1}{c} \int_{ac}^{bc} f\left(\frac{x}{c}\right) \, dx = \int_{a}^{b} f(x) \, dx. \]

### Conclusion

The correct answer is option d:

\[ \int_{a}^{b} f(x) \, dx. \]