Evaluate the integral: [tex]$\int_1^4 \frac{x+2}{\sqrt{x}} dx$[/tex]
Answer (2)
Simplify the integrand: x x + 2 = x 1/2 + 2 x − 1/2 .
Find the antiderivative: ∫ ( x 1/2 + 2 x − 1/2 ) d x = 3 2 x 3/2 + 4 x 1/2 + C .
Evaluate the definite integral: [ 3 2 x 3/2 + 4 x 1/2 ] 1 4 = ( 3 2 ( 4 ) 3/2 + 4 ( 4 ) 1/2 ) − ( 3 2 ( 1 ) 3/2 + 4 ( 1 ) 1/2 ) .
Simplify to find the final answer: 3 26 .
Explanation
Problem Setup We are asked to evaluate the definite integral ∫ 1 4 x x + 2 d x .
Simplifying the Integrand First, we simplify the integrand by dividing each term in the numerator by x : x x + 2 = x x + x 2 = x 1/2 + 2 x − 1/2 .
Finding the Antiderivative Now, we find the antiderivative of x 1/2 + 2 x − 1/2 . Using the power rule for integration, we have ∫ ( x 1/2 + 2 x − 1/2 ) d x = 3/2 x 3/2 + 2 1/2 x 1/2 + C = 3 2 x 3/2 + 4 x 1/2 + C .
Evaluating the Definite Integral Next, we evaluate the definite integral by plugging in the limits of integration: ∫ 1 4 ( x 1/2 + 2 x − 1/2 ) d x = [ 3 2 x 3/2 + 4 x 1/2 ] 1 4 = ( 3 2 ( 4 ) 3/2 + 4 ( 4 ) 1/2 ) − ( 3 2 ( 1 ) 3/2 + 4 ( 1 ) 1/2 ) .
Simplifying the Expression Now, we simplify the expression: ( 3 2 ( 8 ) + 4 ( 2 ) ) − ( 3 2 + 4 ) = 3 16 + 8 − 3 2 − 4 = 3 14 + 4 = 3 14 + 3 12 = 3 26 .
Final Answer Therefore, the value of the definite integral is 3 26 .
Examples
Imagine you're calculating the total amount of water flowing into a reservoir over a period of time, where the rate of flow is given by a function involving a square root. Evaluating the definite integral, as we did here, allows you to determine the total volume of water accumulated between two specific times. This is crucial for managing water resources and planning for irrigation or other needs.
The evaluation of the integral ∫ 1 4 x x + 2 d x simplifies to 3 26 after finding the antiderivative and applying the limits of integration. The process involves breaking down the integrand, finding the antiderivative, and calculating the definite integral. Thus, the final result is 3 26 .
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