Evaluate the Integral

Problem

(∫_1^9)(√(,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) for 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.

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

4. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit of integration and subtracting the value at the lower limit.

[2/3*x(3/2)]91

5. Substitute the upper limit 9 and the lower limit 1 into the expression.

2/3*(9)(3/2)−2/3*(1)(3/2)

6. Calculate the numerical values, noting that 9(3/2)=(√(,9))3=3=27 and 1(3/2)=1

2/3*(27)−2/3*(1)

7. Simplify the final arithmetic to find the value of the definite integral.

18−2/3=54/3−2/3=52/3

Final Answer

(∫_1^9)(√(,x)*d(x))=52/3

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