Find the Antiderivative 2x^(1/2)

Problem

(∫_^)(2*x(1/2)*d(x))

Solution

1. Identify the integral as a power rule problem where the constant multiple can be moved outside the integral.

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

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

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

3. Simplify the exponent and the denominator.

2⋅(x(3/2))/(3/2)+C

4. Multiply by the reciprocal of the denominator to find the final coefficient.

2⋅2/3*x(3/2)+C

4/3*x(3/2)+C

Final Answer

(∫_^)(2*x(1/2)*d(x))=4/3*x(3/2)+C

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