Evaluate the Integral

Problem

(∫_^)(1/√(,8*x)*d(x))

Solution

1. Rewrite the integrand by separating the constant and the variable in the denominator.

1/√(,8*x)=1/(√(,8)⋅√(,x))

2. Simplify the constant √(,8) as 2√(,2) and express the square root of x as a power.

1/(2√(,2)⋅x(1/2))

3. Move the constant outside the integral and rewrite the variable using a negative exponent.

1/(2√(,2))*(∫_^)(x(−1/2)*d(x))

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

1/(2√(,2))⋅(x(1/2))/(1/2)+C

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

1/(2√(,2))⋅2*x(1/2)+C

6. Cancel the common factor of 2 and rewrite the result in radical form.

√(,x)/√(,2)+C

7. Rationalize or combine the radicals into a single square root.

√(,x/2)+C

Final Answer

(∫_^)(1/√(,8*x)*d(x))=√(,2*x)/2+C

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