Evaluate the Integral

Problem

(∫_0^4)(x2*d(x))

Solution

1. Identify the integral as a definite integral of a power function xn where n=2

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

3. Find the antiderivative of x2 which is (x3)/3

4. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit 4 and the lower limit 0

5. Substitute the limits into the expression: 4/3−0/3

6. Simplify the numerical values to find the final result: 64/3−0=64/3

Final Answer

(∫_0^4)(x2*d(x))=64/3

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