# If $\int_1^2 f(x) \ dx = 5$ what can be said about $\int_1^2 3f(\frac{x}{2}) + 1 \ dx$?

This was a question that came up on a short pre-selection exam to enroll in a CS Master degree program.

• a) not enough information
• b) $$16$$
• c) $$31$$
• d) $$\frac{17}{2}$$

I've tried simplifying:

$$\int_1^2 3f(\frac{x}{2}) + 1\ dx$$

Here's what I've done:

1. Extract constant and apply integral addition rule.

$$3\int_1^2 f(\frac{x}{2}) \ dx + \int_1^2 dx$$

1. Compute the second integral of the sum.

$$3\int_1^2 f(\frac{x}{2}) \ dx + x\rvert_1^2 = 3\int_1^2 f(\frac{x}{2}) \ dx + 1$$

I'm not sure how to proceed from here on.

• Are you sure the problem has been copied directly? The first integral involves the values of $f(t)$ when $1<t<2$, while the second integral involves the values of $f(t)$ when $\frac12<t<1$; they have nothing to do with each other. Sep 6, 2023 at 3:42
• To paraphrase @GregMartin, you can say nothing about the second integral. Sep 6, 2023 at 4:01
• In Case the given Integral had limits $(0,1)$ , then the wanted Integral will have limits $(0,0.5)$ & we still can not say much about that , unless we are given a POSITIVE function , which will then make the wanted Integral between $0$ & $16$. When we are then given monotonic function , we might say a little more. Currently , we can say almost nothing.
– Prem
Sep 6, 2023 at 4:19
• @GregMartin I've double-checked, the problem has been copied correctly. It was a multiple-choice answer task and your reply was one of the offered options. So the substitution rule changes the limits of integration. Hence we're looking at a different region under the curve $f (t)$ and we can't say anything about the second integral in relation to the first one. Greg Martin, would you mind promoting your comment to an answer? Sep 6, 2023 at 10:05
• a) is the correct answer as there is no information concerning $f$ on $[1/2,1).$ Sep 6, 2023 at 18:08

The given equation gives us the area under $$f(x)$$ between the lines $$x = 1$$ and $$x=2$$ However, when you look at the second expression, you are asked about the area which is derived from the area under $$f(x)$$ between the lines $$x = \frac{1}{2}$$ and $$x = 1$$ .

now since the function $$f$$ is not defined to us, we can assume any $$f$$ as long as it satisfies the first condition. But 2 different choices of $$f$$ may not give us the same value for the second expression as it covers a different area. Thus we can infer that we have not been given enough information. We can demonstrate this with an example.

• first choice of $$f$$: $$f(x)=5$$

Clearly$$\int_1^2 f(x)dx=5$$, so this equation is valid. according to this choice of function $$f$$, the value of the second expression would be (as you worked out yourself): $$3\times 2.5 +1 = 8.5$$

• second choice of $$f$$: $$f(x) = \frac{10}{3}x$$

again $$\int_1^2 f(x)dx=\left[ \frac53 x^2\right]_1^2 = \frac{20}3 - \frac53=5$$ so this is also valid. according to this, your second expression becomes: $$3\times \left[ \frac53 x^2 \right]_\frac12^1+1$$ $$=3\times \left[ \frac53 - \frac5{12} \right]+1$$ $$=3\times \left[ \frac{15}{12} \right]+1$$ $$=\frac{15}4 +1$$ $$=\frac{19}4$$ $$=4.75$$

As the values of the expressions are different, option $$a$$ would be correct.