Evaluate the Integral

Problem

(∫_1^16)((x−3)/√(,x)*d(x))

Solution

1. Rewrite the integrand by splitting the fraction into two separate terms using the property (a−b)/c=a/c−b/c

(x−3)/√(,x)=x/(x(1/2))−3/(x(1/2))

2. Simplify the terms using the laws of exponents, specifically xa/xb=x(a−b)

x(1−1/2)−3*x(−1/2)=x(1/2)−3*x(−1/2)

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

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

4. Simplify the coefficients of the antiderivative.

2/3*x(3/2)−6*x(1/2)

5. Evaluate the definite integral by plugging in the upper limit 16 and the lower limit 1

(2/3*(16)(3/2)−6*(16)(1/2))−(2/3*(1)(3/2)−6*(1)(1/2))

6. Calculate the numerical values, noting that 16(1/2)=4 and 16(3/2)=4=64

(2/3*(64)−6*(4))−(2/3−6)

7. Combine the terms to find the final result.

(128/3−24)−(2/3−6)

126/3−18

42−18=24

Final Answer

(∫_1^16)((x−3)/√(,x)*d(x))=24

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