Evaluate the Integral

Problem

(∫_2^3)(1/√(,x)−5*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)(5*d(x))

2. Rewrite the first term using exponent notation to prepare for the power rule.

(∫_2^3)(x(−1/2)*d(x))−(∫_2^3)(5*d(x))

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

[2*x(1/2)]32−[5*x]32

4. Simplify the expressions before evaluating at the boundaries.

[2√(,x)]32−[5*x]32

5. Evaluate at the limits by subtracting the value at the lower bound from the value at the upper bound.

(2√(,3)−2√(,2))−(5*(3)−5*(2))

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

2√(,3)−2√(,2)−(15−10)

2√(,3)−2√(,2)−5

Final Answer

(∫_2^3)(1/√(,x)−5*d(x))=2√(,3)−2√(,2)−5

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