Evaluate the Integral

Problem

(∫_^)(3√(,x)*d(x))

Solution

1. Rewrite the radical expression using a fractional exponent to prepare for integration.

√(,x)=x(1/2)

2. Apply the constant multiple rule by moving the constant factor outside the integral.

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

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

3*(∫_^)(x(1/2)*d(x))=3⋅(x(1/2+1))/(1/2+1)+C

4. Simplify the exponent and the denominator.

3⋅(x(3/2))/(3/2)+C

5. Multiply by the reciprocal of the denominator to simplify the expression.

3⋅2/3*x(3/2)+C

6. Cancel the common factor of 3.

2*x(3/2)+C

Final Answer

(∫_^)(3√(,x)*d(x))=2*x(3/2)+C

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