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

Options

A. `(1)/( c) int_(a)^(b) f(x) dx`
B. `int_(a)^(b) f(x) dx`
C. `c int_(a)^(b) f(x) dx`
D. `int_(ac^(2))^(bc^(2)) f(x) 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 integral.

### Step 1: Substitution
Let \( t = \frac{x}{c} \). Then, we differentiate both sides with respect to \( x \):

\[ \frac{dt}{dx} = \frac{1}{c} \implies dx = c \, dt. \]

### Step 2: Change of Limits
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 3: Rewrite the Integral
Substituting \( x \) and \( dx \) in the integral gives us:

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

### Step 4: Factor Out Constants
Now, substituting this back into our original equation, we have:

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

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

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

### Final Result
Thus, we conclude that:

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