first make a substitution and then use integration by parts to evaluate the integral. (use c for the constant of integration.) x ln(5 + x) dx

Step-by-step explanation:

Ok, so we start by setting the integral up. The integral we need to solve is:

so according to the instructions of the problem, we need to start by using some substitution. The substitution will be done as follows:

U=5+x

du=dx

x=U-5

so when substituting the integral will look like this:

now we can go ahead and integrate by parts, remember the integration by parts formula looks like this:

so we must define p, q, p' and q':

p=ln U

q'=U-5

and now we plug these into the formula:

Which simplifies to:

Which solves to:

so we can substitute U back, so we get:

and now we can simplify:

notice how all the constants were combined into one big constant C.

Her method will produce a valid sample.

Her method will produce a random sample.

Her method will produce a representative sample.

Step-by-step explanation: