Evaluate the Integral

Problem

(∫_4^9)(1/√(,x)*d(x))

Solution

1. Rewrite the integrand using a power of x to make it easier to integrate.

1/√(,x)=x(−1/2)

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

(∫_^)(x(−1/2)*d(x))=(x(1/2))/(1/2)

3. Simplify the resulting expression before evaluating the limits.

(x(1/2))/(1/2)=2√(,x)

4. Apply the Fundamental Theorem of Calculus by evaluating the expression at the upper limit of 9 and the lower limit of 4

[2√(,x)]94=2√(,9)−2√(,4)

5. Calculate the final numerical value.

2*(3)−2*(2)=6−4

6−4=2

Final Answer

(∫_4^9)(1/√(,x)*d(x))=2

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