The sought for area A is expressed in terms of the definite integral as

Let

and we get

and

With these results inserted in the integration by parts rule we obtain

The correct answer is E.

This is a Euler equation so we start from the trial solution

Then we get 

and

Substituting these results into the ODE, we obtain the equation in the form

So, the general solution is

The correct answer is C.

Since

and since

Thus upper sum of  corresponding to the given partition is

Thus lower sum of  corresponding to given partition is

The antiderivative of the integrand is

Therefore the correct answer is the B option.

Given parabola intersects the

and

The area of the region corresponding to the interval

equals to the area corresponding to

Hence required area  is calculated by the definite integral

Because we observe that

and

Thus, we get

The right answer is B.

Therefore the correct answer is the E option.

the correct answer is A.

From the linearity of the integral, we find

Observe that

for any arbitrary real number c.

Then the solution can be written as

If we use the initial condition we get that

Thus, we can write the solution as:

Thus, the correct answer is D.

Since 

which is given in the graph, the required answer is found as

Therefore the correct answer is the B option.

As before, we only need to apply the formula for the future value of an income. Since the time period is T = 4 and the interest rate gives r = 0,25, we have

Therefore the correct answer is the B option.

Let 

then

Correct answer is E .

 then du=2xdx or 1/2du=xdx. Substituting these values into the question we have

where c is an arbitrary real number.

We need the intersection points of given curves:

Hence, we have to evaluate the definite integral

on the interval

Therefore required area  is evaluated as

that means the right answer is A.

on the interval

and since the function is even the area A can be calculated by definite integral:

Calculating the integral gives the area as

Hence the correct answer is C.

Hence the correct solution is E.