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

Problem

(∫_^)(2*x^2*d(x))

Solution

1. Identify the function to be integrated, which is ƒ(x)=2*x^2

2. Apply the constant multiple rule for integration, which allows the constant 2 to be moved outside the integral sign.

(∫_^)(2*x^2*d(x))=2*(∫_^)(x^2*d(x))

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

2*(∫_^)(x^2*d(x))=2⋅(x^(2+1))/(2+1)

4. Simplify the expression by performing the addition in the exponent and the denominator.

2⋅(x^3)/3=(2*x^3)/3

5. Add the constant of integration C to represent the family of all possible antiderivatives.

(2*x^3)/3+C

Final Answer

(∫_^)(2*x^2*d(x))=(2*x^3)/3+C

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