Find the Antiderivative 4x-x^2

Problem

(∫_^)(4*x−x2*d(x))

Solution

1. Identify the integral as a sum of two power terms.

2. Apply the sum rule for integration, which allows the integral to be split into two separate integrals.

(∫_^)(4*x−x2*d(x))=(∫_^)(4*x*d(x))−(∫_^)(x2*d(x))

3. Apply the constant multiple rule to the first term by moving the constant outside the integral.

4*(∫_^)(x*d(x))−(∫_^)(x2*d(x))

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

4*((x2)/2)−(x3)/3+C

5. Simplify the coefficients to find the final expression.

2*x2−1/3*x3+C

Final Answer

(∫_^)(4*x−x2*d(x))=2*x2−(x3)/3+C

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