Question

Calc 2 Problem. Could someone help me understand the "u" notation in this integral?

Answer 1

`int_1^0 12usqrt(u^2+1)du=-(8sqrt2-4)=4-8sqrt2`

Explanation:

The two provided integrals look almost exactly the same. In the first, the integrand is in terms of `x,` in the latter, the integrand has all `x` replaced with `u` and is otherwise the same.

The bounds have flipped between the two integrals. Fortunately, there is a rule for handling this:

`int_a^bf(x)dx=-int_b^af(x)dx`

Well, since we know `int_0^1 12xsqrt(x^2+1)dx=8sqrt2-4`, then, for the integral with flipped bounds and a similar integrand, we only have the same answer, but negative:

`int_1^0 12usqrt(u^2+1)du=-(8sqrt2-4)=4-8sqrt2`

Answer 2

The use of `u` is irrelevant.

Explanation:

The definite integral is a number. There is really no variable.

We are given

With `x` as the "dummy variable"

`int_0^1 12xsqrt(x^2+1) dx = 8sqrt2-4`

and also with `t` as the "dummy variable"

`int_0^1 12tsqrt(t^2+1) dt = 8sqrt2-4`

and with `u` as the "dummy variable"

`int_0^1 12usqrt(u^2+1) du = 8sqrt2-4`

and so on.

The question asks us to find

`int_1^0 12usqrt(u^2+1) du`

which is exactly the same as

`int_1^0 12xsqrt(x^2+1) dx`

and

`int_1^0 12tsqrt(t^2+1) dt`

and

`int_1^0 12rsqrt(r^2+1) dr`

As the answer by VNVDVI says, the integral is `-(8sqrt2-4) = 4-8sqrt2`.