Given that `int_0^1(15xsqrt(x^2 + 9))dx=50sqrt(10)-135` , what is `int_1^0(15usqrt(u^2 + 9))du` ?

We are given
that `int_(0)^(1)(15xsqrt(x^2+9))dx=50sqrt(10)-135` and we are asked to evaluate
`int_(1)^(0)(15usqrt(u^2+9))du` .

These are definite integrals. If we
restrict the variables to the real number line, a definite integral is typically defined as the
Riemann integral; i.e. an infinite sum of geometric shapes. Definite integrals can be described
as a "signed" area.

By the Fundamental Theorem of Calculus, we can
find the definite integral by integrating the indefinite integral and evaluating at the
endpoints. `int_(a)^(b)f(x)dx=F(b)-F(a)` where F is the function whose derivative is
f.

So one way to evaluate the given integral is to integrate the indefinite
integral. Thus

`int(15usqrt(u^2+9))du=15int(u(u^2+9))du`

Then noticing that d/du of u^2+9 is 2udu we can multiply inside the integral by 2 and
outside by 1/2 to get:

`=15/2int(2u(u^2+9)^(1/2))du`

`=15/2 * ((u^2+9)^(3/2))/(3/2)+C`

`=5(u^2+9)^(3/2)+C`

The definite integral is

`5(u^2+9)^(3/2)|_(1)^(0)`

`=(5*9^(3/2))-(5*(10)^(3/2))`

`=135-50sqrt(10)`

But there is an easier way. There is a rule for integrals that states:

`int_a^b f(x)dx=-int_b^af(x)dx`

So we take the given
value of the indefinite integral and multiply by negative one to get

`135-50sqrt(10)`

The functions in the integrand are identical
except for the variable. The only other difference between the two definite integrals is that
the upper and lower limits of integration have been switched. Thus we can use the rule giving us
the answer of the opposite of the original definite integral.

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