Evaluate the following [tex]$\int \sqrt{x} dx$[/tex].
Answer (1)
Rewrite the integrand: x = x 2 1 .
Apply the power rule for integration: ∫ x n d x = n + 1 x n + 1 + C .
Calculate the integral: ∫ x 2 1 d x = 2 3 x 2 3 + C .
Simplify the result: 3 2 x 2 3 + C .
Explanation
Problem Analysis We are asked to evaluate the indefinite integral of x with respect to x . This means we need to find a function whose derivative is x .
Applying the Power Rule We can rewrite x as x 2 1 . Now we can use the power rule for integration, which states that ∫ x n d x = n + 1 x n + 1 + C , where n is any real number except − 1 , and C is the constant of integration.
Calculating the Integral In our case, n = 2 1 . So, we have n + 1 = 2 1 + 1 = 2 3 . Therefore, the integral becomes: ∫ x 2 1 d x = 2 3 x 2 3 + C
Simplifying the Result Simplifying the expression, we get: 2 3 x 2 3 + C = 3 2 x 2 3 + C
Final Answer Thus, the indefinite integral of x is 3 2 x 2 3 + C .
Examples
Imagine you're calculating the total amount of water collected in a reservoir over time, where the rate of water flow into the reservoir is proportional to the square root of time. The integral of the square root function, as we just calculated, would help you determine the total volume of water accumulated at any given time. This kind of calculation is crucial for managing water resources and planning for future needs.
