Evaluate the Integral

Problem

(∫_^)((6*x^2−2*x−7)/√(,x)*d(x))

Solution

1. Rewrite the integrand by expressing the square root as a power and splitting the fraction into three separate terms.

(∫_^)(((6*x^2)/(x^(1/2))−(2*x)/(x^(1/2))−7/(x^(1/2)))*d(x))

2. Simplify each term using the laws of exponents, specifically x^a/x^b=x^(a−b)

(∫_^)((6*x^(3/2)−2*x^(1/2)−7*x^(−1/2))*d(x))

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

6*((x^(5/2))/(5/2))−2*((x^(3/2))/(3/2))−7*((x^(1/2))/(1/2))+C

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

12/5*x^(5/2)−4/3*x^(3/2)−14*x^(1/2)+C

Final Answer

(∫_^)((6*x^2−2*x−7)/√(,x)*d(x))=12/5*x^(5/2)−4/3*x^(3/2)−14*x^(1/2)+C

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