Evaluate the Integral

Problem

(∫_0^1)(x2*d(x))

Solution

1. Identify the integrand and the limits of integration. The function to integrate is ƒ(x)=x2 and the interval is [0,1]

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

3. Find the antiderivative of x2 by increasing the exponent by 1 and dividing by the new exponent.

(∫_^)(x2*d(x))=(x3)/3

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

[(x3)/3]10=1/3−0/3

5. Simplify the resulting numerical expression.

1/3−0=1/3

Final Answer

(∫_0^1)(x2*d(x))=1/3

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