Evaluate the Integral integral of 25-x^2 with respect to x

Problem

(∫_^)(25−x2*d(x))

Solution

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

(∫_^)(25−x2*d(x))=(∫_^)(25*d(x))−(∫_^)(x2*d(x))

2. Integrate the constant term using the rule (∫_^)(a*d(x))=a*x

(∫_^)(25*d(x))=25*x

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

(∫_^)(x2*d(x))=(x3)/3

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

25*x−(x3)/3+C

Final Answer

(∫_^)(25−x2*d(x))=25*x−(x3)/3+C

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