Final answer:
By evaluating the integral ∫20dx2−x−−−−−√), we get 2(2(³/²)), thus the correct option is B.
Step-by-step explanation:
To evaluate the given integral, we will first rewrite it in a form that is easier to work with.
Using the exponent properties, we can rewrite the integral as ∫20dx(2-x)-½. Now, we can use the power rule of integration, which states that ∫xndx = x(n+1)/(n+1) + C, to solve the integral.
∫20dx(2-x)-½ = (2-x)½ + C
Next, we will apply the limits of integration, which in this case are from 0 to 2. So, our final answer will be [(2-2)½ - (2-0)½] = (2½ - 0½) = (2(³/²)) = B.
In the given integral, we have been asked to evaluate ∫20dx2−x−−−−−√. To solve this integral, we first need to rewrite it in a form that is easier to work with. Using the exponent properties, we can rewrite the integral as ∫20dx(2-x)-½.
This form will allow us to apply the power rule of integration, which states that ∫xndx = x(n+1)/(n+1) + C.
Now, we can apply the power rule to our integral and solve it as follows:
∫20dx(2-x)-½ = (2-x)½ + C
Next, we will apply the limits of integration, which in this case are from 0 to 2. So, our final answer will be [(2-2)½ - (2-0)½] = (2½ - 0½) = (2(³/²)) = B. This is the final answer to the given integral.
In the explanation part, we have used the power rule of integration to solve the given integral. This rule states that if we have an integral of the form ∫xndx, we can solve it by using the formula x(n+1)/(n+1) + C. In our case, we had an additional term of (2-x)-½, which we could rewrite as (2-x)½. By applying the power rule, we were able to simplify the integral to (2-x)½ + C.
Next, we applied the limits of integration to our integral, which were from 0 to 2. This means that we have to evaluate the integral at x=2 and x=0 and then take the difference between the two values. So, our final answer became [(2-2)½ - (2-0)½] = (2½ - 0½) = (2(³/²)) = B.
In conclusion, by following the instructions accurately and using the power rule of integration, we were able to solve the given integral and arrive at the final answer of B) 2(2(³/²)). It is important to carefully rewrite the integral in a form that is easier to work with and then apply the appropriate integration rules. By dividing the explanation into different paragraphs, we have made the solution easier to understand and follow.