Evaluate the Integral

Problem

(∫_1^16)((x−5)/√(,x)*d(x))

Solution

1. Rewrite the integrand by splitting the fraction into two separate terms.

(x−5)/√(,x)=x/(x^(1/2))−5/(x^(1/2))

2. Simplify the exponents using the rule (x^a)/(x^b)=x^(a−b)

x^(1−1/2)−5*x^(−1/2)=x^(1/2)−5*x^(−1/2)

3. Apply the power rule for integration, (∫_^)(x^n*d(x))=(x^(n+1))/(n+1) to each term.

(∫_^)((x^(1/2)−5*x^(−1/2))*d(x))=(x^(3/2))/(3/2)−(5*x^(1/2))/(1/2)

4. Simplify the coefficients of the antiderivative.

2/3*x^(3/2)−10*x^(1/2)

5. Evaluate the definite integral at the upper limit x=16

2/3*(16)^(3/2)−10*(16)^(1/2)=2/3*(64)−10*(4)

128/3−40=128/3−120/3=8/3

6. Evaluate the definite integral at the lower limit x=1

2/3*(1)^(3/2)−10*(1)^(1/2)=2/3−10

2/3−30/3=−28/3

7. Subtract the lower limit value from the upper limit value.

8/3−(−28/3)=8/3+28/3

36/3=12

Final Answer

(∫_1^16)((x−5)/√(,x)*d(x))=12

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