Evaluate the Integral

Problem

(∫_0^27)(√(,3*x)*d(x))

Solution

1. Rewrite the integrand using exponent notation and the property of radicals √(,a*b)=√(,a)√(,b)

(∫_0^27)(√(,3)⋅x(1/2)*d(x))

2. Factor out the constant √(,3) from the integral.

√(,3)*(∫_0^27)(x(1/2)*d(x))

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

√(,3)*[(x(3/2))/(3/2)]270

4. Simplify the constant coefficient by multiplying by the reciprocal.

(2√(,3))/3*[x(3/2)]270

5. Evaluate the expression at the upper and lower limits of integration.

(2√(,3))/3*(27(3/2)−0(3/2))

6. Calculate the value of 27(3/2) by taking the square root of 27 and cubing it, or noting 27=3

27(3/2)=(√(,27))3=(3√(,3))3=27⋅3√(,3)=81√(,3)

7. Multiply the resulting values to find the final numerical answer.

(2√(,3))/3⋅81√(,3)=(162⋅3)/3=162

Final Answer

(∫_0^27)(√(,3*x)*d(x))=162

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