Evaluate the Integral

Problem

(∫_^)(√(,2*x)*d(x))

Solution

1. Rewrite the integrand using exponent notation to make it easier to apply the power rule.

(∫_^)(√(,2)⋅x(1/2)*d(x))

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

√(,2)*(∫_^)(x(1/2)*d(x))

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

√(,2)⋅(x(1/2+1))/(1/2+1)+C

4. Simplify the exponent and the denominator.

√(,2)⋅(x(3/2))/(3/2)+C

5. Multiply by the reciprocal of the denominator and combine terms.

√(,2)⋅2/3*x(3/2)+C

6. Finalize the expression by writing it in a standard form.

(2√(,2))/3*x(3/2)+C

Final Answer

(∫_^)(√(,2*x)*d(x))=(2√(,2))/3*x(3/2)+C

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