Evaluate the Integral integral of 6x^(1/2) with respect to x

Problem

(∫_^)(6*x(1/2)*d(x))

Solution

1. Identify the constant and the power of x in the integrand. The constant is 6 and the exponent is n=1/2

2. Apply the constant multiple rule for integration by moving the 6 outside the integral.

(∫_^)(6*x(1/2)*d(x))=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 for n≠−1

6*(∫_^)(x(1/2)*d(x))=6⋅(x(1/2+1))/(1/2+1)+C

4. Simplify the exponent and the denominator.

6⋅(x(3/2))/(3/2)+C

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

6⋅2/3*x(3/2)+C

6. Calculate the final coefficient.

4*x(3/2)+C

Final Answer

(∫_^)(6*x(1/2)*d(x))=4*x(3/2)+C

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