Find the Antiderivative f(x)=x(2-x)^2

Problem

(∫_^)(x*(2−x)^2*d(x))

Solution

1. Expand the squared binomial (2−x)^2 using the formula (a−b)^2=a^2−2*a*b+b^2

(2−x)^2=4−4*x+x^2

2. Distribute the x into the expanded polynomial.

x*(4−4*x+x^2)=4*x−4*x^2+x^3

3. Apply the sum rule for integration to integrate each term separately.

(∫_^)((4*x−4*x^2+x^3)*d(x))=(∫_^)(4*x*d(x))−(∫_^)(4*x^2*d(x))+(∫_^)(x^3*d(x))

4. Use the power rule for integration, (∫_^)(x^n*d(x))=(x^(n+1))/(n+1) for each term.

(∫_^)(4*x*d(x))=(4*x^2)/2=2*x^2

(∫_^)(4*x^2*d(x))=(4*x^3)/3

(∫_^)(x^3*d(x))=(x^4)/4

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

2*x^2−(4*x^3)/3+(x^4)/4+C

Final Answer

(∫_^)(x*(2−x)^2*d(x))=2*x^2−(4*x^3)/3+(x^4)/4+C

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