[tex]$\int_1^4\left(\frac{k+2}{\sqrt{k}}\right) d k$[/tex]
Answer (1)
Rewrite the integrand: k k + 2 = k 1/2 + 2 k − 1/2 .
Find the antiderivative: ∫ ( k 1/2 + 2 k − 1/2 ) d k = 3 2 k 3/2 + 4 k 1/2 + C .
Evaluate the definite integral: [ 3 2 k 3/2 + 4 k 1/2 ] 1 4 = ( 3 2 ( 4 ) 3/2 + 4 ( 4 ) 1/2 ) − ( 3 2 ( 1 ) 3/2 + 4 ( 1 ) 1/2 ) .
Simplify to get the final answer: 3 26 .
Explanation
Problem Setup We are asked to evaluate the definite integral ∫ 1 4 ( k k + 2 ) d k .
Rewrite the Integrand First, we rewrite the integrand as a sum of powers of k :
k k + 2 = k k + k 2 = k 1/2 + 2 k − 1/2 .
Find the Antiderivative Next, we find the antiderivative of k 1/2 + 2 k − 1/2 with respect to k . Recall that the power rule for integration states that ∫ x n d x = n + 1 x n + 1 + C for n = − 1 . Thus, ∫ ( k 1/2 + 2 k − 1/2 ) d k = 3/2 k 3/2 + 2 1/2 k 1/2 + C = 3 2 k 3/2 + 4 k 1/2 + C .
Evaluate the Definite Integral Now, we evaluate the definite integral by finding the difference of the antiderivative at the upper and lower limits of integration: ∫ 1 4 ( k 1/2 + 2 k − 1/2 ) d k = [ 3 2 k 3/2 + 4 k 1/2 ] 1 4 = ( 3 2 ( 4 ) 3/2 + 4 ( 4 ) 1/2 ) − ( 3 2 ( 1 ) 3/2 + 4 ( 1 ) 1/2 ) .
Simplify the Expression 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 designing a water fountain where the water flow rate is described by the function k k + 2 . Evaluating the definite integral ∫ 1 4 ( k k + 2 ) d k would help you calculate the total amount of water that flows from the fountain between time k = 1 and k = 4 . This is crucial for determining the size of the water reservoir needed or for optimizing the fountain's water usage.
