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

Problem

(∫_^)((x+1)*(2*x−1)*d(x))

Solution

1. Expand the binomial expression using the FOIL method to make it easier to integrate.

(x+1)*(2*x−1)=2*x2−x+2*x−1

2. Simplify the resulting polynomial by combining like terms.

2*x2+x−1

3. Apply the sum rule for integrals, which allows for the integration of each term individually.

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

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

(2*x3)/3+(x2)/2−x+C

Final Answer

(∫_^)((x+1)*(2*x−1)*d(x))=(2*x3)/3+(x2)/2−x+C

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