Evaluate the Integral

Problem

(∫_4^9)(√(,x)*d(x))

Solution

1. Rewrite the integrand using a fractional exponent to make it easier to apply the power rule for integration.

√(,x)=x(1/2)

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

(∫_^)(x(1/2)*d(x))=(x(3/2))/(3/2)

3. Simplify the resulting expression by multiplying by the reciprocal of the denominator.

(2*x(3/2))/3

4. Evaluate the definite integral at the upper limit of 9 and the lower limit of 4

[(2*x(3/2))/3]94=(2*(9)(3/2))/3−(2*(4)(3/2))/3

5. Calculate the numerical values by taking the square root of the bases first and then cubing them.

(2*(3))/3−(2*(2))/3

6. Simplify the fractions to find the final difference.

54/3−16/3=38/3

Final Answer

(∫_4^9)(√(,x)*d(x))=38/3

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