How to compute $\int_0^2 f(4x)\, dx$ given $\int_0^8 f(x)\, dx=4$

Question

Compute $\displaystyle \int_0^2 f(4x)\, dx$ given that $\displaystyle\int_0^8 f(x)\, dx=4$.

At first I thought this was an ‘integrate by recognition’ type of question, I but can’t seem to come up with an answer.

Can someone tell me what sort of method I should use?

Answer 1

Use the substitution $u = 4x$:

$$u = 4x \quad \mathrm du = 4\,\mathrm dx$$

From this, our new bounds are $u = 4 \cdot 0$ and $u = 4 \cdot 2$:

$$\frac 14 \int_0^8 f(u)\,\mathrm du = \frac 14 \cdot 4 = 1$$

We can conclude that:

$$\int_0^2 f(4x)\,\mathrm dx = 1$$

Answer 2

Let $u=4x$, then you get $$\int_0^2 f(4x)\, dx= (1/4)\int_0^8 f(u)\, du=1$$

Answer 3

We have , $\int_{0}^{8} f(u)du = 4$ . Change of variable doesnot affect the integration .

Now , let $\ x = \frac{u}{4}$ , then $\ dx = \frac{du}{4}$ and when$\ u = 0$ then,$\ x = 0$ and when$\ u = 8$ then$\ x = 2$ .

So, we have $\int_{0}^8 f(u)du = 4\int_{0}^2 f(4x)dx = 4$ , i.e $\int_{0}^2 f(4x)dx = 1$ .