Find the Antiderivative f(x)=1/2x^2-2x+6

Problem

(∫_^)(1/2*x2−2*x+6*d(x))

Solution

1. Identify the integral of the function by applying the sum rule for integration, which allows the integration of each term individually.

(∫_^)((1/2*x2−2*x+6)*d(x))=(∫_^)(1/2*x2*d(x))−(∫_^)(2*x*d(x))+(∫_^)(6*d(x))

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

(∫_^)(1/2*x2*d(x))=1/2⋅(x3)/3=(x3)/6

3. Integrate the linear term using the power rule where the exponent is 1.

(∫_^)(2*x*d(x))=2⋅(x2)/2=x2

4. Integrate the constant term by appending the variable x

(∫_^)(6*d(x))=6*x

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

(x3)/6−x2+6*x+C

Final Answer

(∫_^)(1/2*x2−2*x+6*d(x))=1/6*x3−x2+6*x+C

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