Evaluate the Integral

Problem

(∫_^)(12/√(,x)*d(x))

Solution

1. Rewrite the integrand using exponent notation by expressing the square root as a power and moving it to the numerator.

(∫_^)(12*x(−1/2)*d(x))

2. Apply the constant multiple rule for integration to move the coefficient outside the integral.

12*(∫_^)(x(−1/2)*d(x))

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

12⋅(x(−1/2+1))/(−1/2+1)+C

4. Simplify the exponent and the denominator.

12⋅(x(1/2))/(1/2)+C

5. Multiply by the reciprocal of the denominator and convert the fractional exponent back to radical form.

24√(,x)+C

Final Answer

(∫_^)(12/√(,x)*d(x))=24√(,x)+C

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