Evaluate the Integral

Problem

(∫_0^8)(√(,8*x)*d(x))

Solution

1. Rewrite the integrand using the properties of radicals to separate the constant and the variable.

√(,8*x)=√(,8)⋅√(,x)

2. Simplify the constant √(,8) and rewrite the variable part using a fractional exponent.

√(,8)=2√(,2)

√(,x)=x(1/2)

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

(∫_^)(2√(,2)*x(1/2)*d(x))=2√(,2)⋅(x(3/2))/(3/2)

4. Simplify the expression by multiplying by the reciprocal of the denominator.

2√(,2)⋅2/3*x(3/2)=(4√(,2))/3*x(3/2)

5. Evaluate the definite integral at the upper limit of 8 and the lower limit of 0

(4√(,2))/3*(8)(3/2)−(4√(,2))/3*(0)(3/2)

6. Calculate the value of 8(3/2) by taking the square root of 8 and raising it to the third power.

8(3/2)=(√(,8))3=(2√(,2))3

(2√(,2))3=8⋅2√(,2)=16√(,2)

7. Multiply the results to find the final numerical value.

(4√(,2))/3⋅16√(,2)=(64⋅2)/3

128/3

Final Answer

(∫_0^8)(√(,8*x)*d(x))=128/3

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