Evaluate the Integral

Problem

(∫_0^2)(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 [0,2]

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 and subtracting the evaluation at the lower limit.

[(3*x2)/2]20

5. Substitute the upper limit x=2

(3*(2)2)/2=(3*(4))/2=6

6. Substitute the lower limit x=0

(3*(0)2)/2=0

7. Subtract the results to find the final value.

6−0=6

Final Answer

(∫_0^2)(3*x*d(x))=6

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