Find the Antiderivative f(x)=x^(1/2)

Problem

(∫_^)(x^(1/2)*d(x))

Solution

1. Identify the function to be integrated, which is a power function of the form x^n where n=1/2

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

3. Add one to the exponent: $1/2 + 1 = 3/2$.

4. Divide by the new exponent: (x^(3/2))/(3/2)

5. Simplify the fraction by multiplying by the reciprocal of the denominator: 2/3*x^(3/2)

6. Add the constant of integration C to represent the family of all antiderivatives.

Final Answer

(∫_^)(x^(1/2)*d(x))=2/3*x^(3/2)+C

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