Evaluate the Integral

Problem

(∫_^)((x+4)/√(,x)*d(x))

Solution

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

(∫_^)(x/√(,x)+4/√(,x)*d(x))

2. Simplify the terms using the properties of exponents, where √(,x)=x^(1/2)

(∫_^)(x^(1/2)+4*x^(−1/2)*d(x))

3. Apply the power rule for integration, which states (∫_^)(x^n*d(x))=(x^(n+1))/(n+1)+C

(x^(3/2))/(3/2)+(4*x^(1/2))/(1/2)+C

4. Simplify the coefficients by multiplying by the reciprocals of the denominators.

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

Final Answer

(∫_^)((x+4)/√(,x)*d(x))=2/3*x^(3/2)+8*x^(1/2)+C

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