Evaluate the Integral

Problem

(∫_0^3)(3*x*d(x))

Solution

1. Identify the integrand and the limits of integration. The function to integrate is ƒ(x)=3*x and the interval is [0,3]

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

3. Find the antiderivative of 3*x

(∫_^)(3*x*d(x))=(3*x2)/2

4. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit 3 and subtracting the value at the lower limit 0

[(3*x2)/2]30

5. Substitute the values into the expression.

(3*(3)2)/2−(3*(0)2)/2

6. Simplify the arithmetic to find the final value.

(3*(9))/2−0=27/2

Final Answer

(∫_0^3)(3*x*d(x))=27/2

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