Question

$\int_{a/c}^{b/c} f(x)dx =$

Answer 1

$\frac{1}{c}\int_{a}^{b} f\left(\frac{x}{c}\right)dx$

Answer 2

To evaluate the definite integral $\int_{a/c}^{b/c} f(x)dx$, we can use the method of substitution.

Let's make a substitution that transforms the limits of integration from $a/c$ and $b/c$ to $a$ and $b$. Let $x = \frac{u}{c}$.

Now, we need to find $dx$ in terms of $du$: Differentiating both sides with respect to $u$: $\frac{dx}{du} = \frac{1}{c}$ So, $dx = \frac{1}{c} du$.

Next, we need to change the limits of integration according to the new variable $u$: When the lower limit $x = a/c$: Substitute into $x = u/c \Rightarrow a/c = u/c \Rightarrow u = a$.

When the upper limit $x = b/c$: Substitute into $x = u/c \Rightarrow b/c = u/c \Rightarrow u = b$.

Now, substitute $x = u/c$ and $dx = \frac{1}{c} du$ into the integral, along with the new limits: $\int_{a/c}^{b/c} f(x)dx = \int_{a}^{b} f\left(\frac{u}{c}\right) \frac{1}{c} du$ Since $\frac{1}{c}$ is a constant, we can take it out of the integral: $= \frac{1}{c} \int_{a}^{b} f\left(\frac{u}{c}\right) du$ The variable of integration is a dummy variable, so we can replace $u$ with $x$ without changing the value of the definite integral: $= \frac{1}{c} \int_{a}^{b} f\left(\frac{x}{c}\right) dx$