Evaluate the Integral

Problem

(∫_^)((6*x2−2*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*x2)/(x(1/2))−(2*x)/(x(1/2))−7/(x(1/2)))*d(x))

2. Simplify each term using the laws of exponents, specifically xa/xb=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 (∫_^)(xn*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*x2−2*x−7)/√(,x)*d(x))=12/5*x(5/2)−4/3*x(3/2)−14*x(1/2)+C

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