Evaluate the Integral

Problem

(∫_1^4)(3*x*d(x))

Solution

1. Identify the integrand and the limits of integration. The function is 3*x and we are integrating from x=1 to x=4

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

(∫_^)(3*x*d(x))=(3*x2)/2

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

[(3*x2)/2]41

4. Substitute the upper limit x=4 into the expression.

(3*(4)2)/2=(3*(16))/2=24

5. Substitute the lower limit x=1 into the expression.

(3*(1)2)/2=3/2

6. Subtract the lower limit value from the upper limit value to find the final result.

24−3/2=48/2−3/2=45/2

Final Answer

(∫_1^4)(3*x*d(x))=45/2

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