Evaluate the Integral

Problem

(∫_^)(√(,6*x)*d(x))

Solution

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

(∫_^)(√(,6)⋅√(,x)*d(x))

(∫_^)(√(,6)⋅x(1/2)*d(x))

2. Factor out the constant √(,6) from the integral.

√(,6)*(∫_^)(x(1/2)*d(x))

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

√(,6)⋅(x(1/2+1))/(1/2+1)+C

√(,6)⋅(x(3/2))/(3/2)+C

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

√(,6)⋅2/3*x(3/2)+C

(2√(,6))/3*x(3/2)+C

Final Answer

(∫_^)(√(,6*x)*d(x))=(2√(,6))/3*x(3/2)+C

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