Find the Antiderivative

Problem

(∫_^)((1/2*x(1/3)−2/(x3)+3/√(,x))*d(x))

Solution

1. Rewrite the expression using power notation to make it easier to apply the power rule for integration.

1/2*x(1/3)−2*x(−3)+3*x(−1/2)

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

(∫_^)(1/2*x(1/3)*d(x))=1/2⋅(x(4/3))/(4/3)=1/2⋅3/4*x(4/3)=3/8*x(4/3)

3. Integrate the second term using the same power rule.

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

4. Integrate the third term by adding 1 to the exponent and dividing by the new exponent.

(∫_^)(3*x(−1/2)*d(x))=3⋅(x(1/2))/(1/2)=3⋅2*x(1/2)=6*x(1/2)

5. Combine the results and add the constant of integration C

3/8*x(4/3)+x(−2)+6*x(1/2)+C

Final Answer

(∫_^)((1/2*x(1/3)−2/(x3)+3/√(,x))*d(x))=3/8*x(4/3)+1/(x2)+6√(,x)+C

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