Evaluate the following integral: [tex]$\int 3 \sqrt{x} dx$[/tex]
Answer (1)
Rewrite the integral: 3 ∫ x d x = 3 ∫ x 2 1 d x .
Apply the power rule for integration: ∫ x n d x = n + 1 x n + 1 + C .
Calculate the integral: 3 ⋅ 2 3 x 2 3 + C = 2 x 2 3 + C .
The indefinite integral is: 2 x 2 3 + C .
Explanation
Problem Analysis We are asked to evaluate the indefinite integral ∫ 3 x d x . This means we need to find a function whose derivative is 3 x .
Rewriting the Integral First, we can rewrite the integral as 3 ∫ x d x = 3 ∫ x 2 1 d x . This makes it easier to apply the power rule for integration.
Applying the Power Rule The power rule for integration states that ∫ x n d x = n + 1 x n + 1 + C , where n is a constant and C is the constant of integration. In our case, n = 2 1 .
Calculating the Integral Applying the power rule, we have:
∫ x 2 1 d x = 2 1 + 1 x 2 1 + 1 + C = 2 3 x 2 3 + C = 3 2 x 2 3 + C
Multiplying by the Constant Now, we multiply by the constant 3 that we factored out earlier:
3 ∫ x 2 1 d x = 3 ⋅ 3 2 x 2 3 + C = 2 x 2 3 + C
Final Result Therefore, the indefinite integral of 3 x is 2 x 2 3 + C .
Examples
Imagine you are calculating the total production of a factory where the production rate increases with the square root of time. The integral of 3 x (where x is time) would give you the total units produced over a period. This type of calculation is crucial for planning and resource allocation in manufacturing and other industries.
