Find the Antiderivative f(x)=5x^2-5x

Problem

(∫_^)(5*x2−5*x*d(x))

Solution

1. Identify the integral to be solved, which represents the antiderivative of the function ƒ(x)=5*x2−5*x

(∫_^)(5*x2−5*x*d(x))

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

(∫_^)(5*x2*d(x))−(∫_^)(5*x*d(x))

3. Apply the constant multiple rule to move the coefficients outside of the integrals.

5*(∫_^)(x2*d(x))−5*(∫_^)(x1*d(x))

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

5(x3)/3−5(x2)/2+C

5. Simplify the expression to find the final form of the antiderivative.

5/3*x3−5/2*x2+C

Final Answer

(∫_^)(5*x2−5*x*d(x))=5/3*x3−5/2*x2+C

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