Evaluate the Integral

Problem

(∫_2^3)(1/√(,x)−7*d(x))

Solution

1. Split the integral into two separate terms using the linearity property of integration.

(∫_2^3)(1/√(,x)*d(x))−(∫_2^3)(7*d(x))

2. Rewrite the first term using exponent notation to make it easier to apply the power rule.

(∫_2^3)(x(−1/2)*d(x))−(∫_2^3)(7*d(x))

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

[2*x(1/2)]32−[7*x]32

4. Simplify the expressions before evaluating the limits of integration.

[2√(,x)]32−[7*x]32

5. Evaluate at the boundaries by subtracting the value at the lower limit from the value at the upper limit for both terms.

(2√(,3)−2√(,2))−(7*(3)−7*(2))

6. Simplify the constants to find the final numerical expression.

2√(,3)−2√(,2)−(21−14)

2√(,3)−2√(,2)−7

Final Answer

(∫_2^3)(1/√(,x)−7*d(x))=2√(,3)−2√(,2)−7

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