Evaluate the Integral

Problem

(∫_^)(12/√(,x)+12√(,x)*d(x))

Solution

1. Rewrite the integrand using power notation to make it easier to apply the power rule for integration.

12/√(,x)+12√(,x)=12*x(−1/2)+12*x(1/2)

2. Apply the sum rule for integrals to integrate each term separately.

(∫_^)(12*x(−1/2)+12*x(1/2)*d(x))=(∫_^)(12*x(−1/2)*d(x))+(∫_^)(12*x(1/2)*d(x))

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

(∫_^)(12*x(−1/2)*d(x))=12⋅(x(1/2))/(1/2)=24*x(1/2)

(∫_^)(12*x(1/2)*d(x))=12⋅(x(3/2))/(3/2)=8*x(3/2)

4. Combine the results and add the constant of integration C

24*x(1/2)+8*x(3/2)+C

5. Simplify the expression by converting the fractional exponents back into radical form.

24√(,x)+8√(,x3)+C

Final Answer

(∫_^)(12/√(,x)+12√(,x)*d(x))=24√(,x)+8*x√(,x)+C

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