Evaluate the Integral

Problem

(∫_1^3)(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 [1,3]

2. Apply the power rule for integration, which states that (∫_^)(xn*d(x))=(x(n+1))/(n+1) For n=2 the antiderivative is (x3)/3

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

F(x)=(x3)/3

(∫_1^3)(x2*d(x))=F(3)−F(1)

4. Substitute the values into the expression.

(∫_1^3)(x2*d(x))=3/3−1/3

5. Simplify the numerical result.

(∫_1^3)(x2*d(x))=27/3−1/3

(∫_1^3)(x2*d(x))=26/3

Final Answer

(∫_1^3)(x2*d(x))=26/3

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