A. one way to integrate ∫x√x+1 dx is to use integration by parts. do so to find the antiderivative.
b. another way to evaluate the integral in part a is by using the substitution u=x+1. do so to find the antiderivative.
c. compare the results from the two methods. if they do not look the same, explain how this can happen. discuss the advantages and disadvantages of each method.

By parts:

By substitution:

As you can see, no difference (as long as you simplify everything in a consistent way).

infinitely many solutions

step-by-step explanation:

we have been given the equation;

2(2x + 3) = -4 + 2(2x + 5)

we need to determine the value of x.

2(2x + 3), we expand this expression to obtain 4x + 6

similarly 2(2x + 5) will become 4x + 10

now we have;

4x + 6 = -4 + 4x + 10

4x + 6 = 4x + 10 - 4

4x + 6 = 4x + 6

the expression on the left hand side is identical to the one on the right hand side. therefore, we are looking at just a single straight line whose equation is;

y = 4x + 6

consequently, we shall have infinitely many solutions