If this integral equals this, then what does this integral equal?

Question

Find the integral from 0 to 1 of f(2x+1) dx
If the integral from 1 to 2 of f(x) dx = 4

Thanks for help, it's greatly appreciated as always! :)

sorry the second integral should read from 1 to 3 ... ARGH. hopefully this makes it easier to solve?

looks like you guys got the answers though. thanks!

Answer 1

Sorry, this question does not have a definite answer. Are you sure you did not make a mistake in typing it in?

Integral (0 to 1) of f(2x + 1) dx: let u = 2x+1, du = 2dx
= Integral (1 to 3) of f(u) / 2 du
= (1/2) . Integral (1 to 3) of f(x) dx.

So, if it was supposed to be that the integral from 1 to 3 of f(x) dx is 4, then the answer is 2. Otherwise, the best we can say is that it is 2 + half of the integral from 2 to 3 of f(x) dx.

Answer 2

These ones are always confusing until you look beyond the little annoyance.

To solve this, you have to use substitution.
Let u = 2x + 1
Then du = 2 dx, and (1/2)du = dx
When doing this, remember that our bounds for integration change as well.
when x = 0, u = 1
when x = 1, u = 3

So our new integral began as

Integral (0 to 1, f(2x+1))dx

And ends up as

Integral (1 to 3, f(u) (1/2) ) du

We can pull out the constant 1/2, to get

(1/2) * Integral (1 to 3, f(u)) du

Note that this is EXACTLY the same as what you wrote as the second integral, Integral (1 to 3, f(x) dx), but with the variable u instead of x. This is NOT relevant at all. That is,

Integral (1 to 3, f(u)) du = Integral (1 to 3, f(x)) dx, which equals 4.

Therefore,

(1/2) * Integral (1 to 3, f(u)) du = (1/2) [4] = 4/2 = 2

The key thing is to look beyond the fact that you're looking at the integral in terms of u, even though what you're given is expressed in terms of x.

Answer 3

Substitute u = 2x + 1. Then dx/du = 1/2, and the integral from x = 0 to x = 1 of f(2x + 1) dx is equal to
int f(u) (1/2) du, from u = 1 to u = 3, i.e. it is

1/2 * int [1 to 3] of f(u) du. Now, in a definite integral, the variable is a dummy, so this is the same as
1/2 * int [1 to 3] of f(x) dx. I think we have a misprint here, I suspect the question should give int 1 to 3, not 1 to 2, as being equal to 4, and then you have the answer.

Answer 4

Let F'(x) = f(x)

Therefore, using the Fundamental Theorem of Calculus,
∫(0,1) f(2x + 1) dx
= F(2x + 1)|(0,1)
= F[2(1) + 1] - F[2(0) + 1]
= F(2 + 1) - F(0 + 1)
= F(3) - F(1)

Now, we know that
∫(1,2) f(x)dx = 4

Thus,
F(x)|(1,2) = 4

Then,
F(2) - F(1) = 4

Here, we cannot find the integral because we are only given the integral from 1 to 2. The integral from 2 to 3 is not given.

^_^

Answer 5

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