Evaluate the Integral

Problem

(∫_1^2)(4*x3*d(x))

Solution

1. Identify the integrand and the limits of integration. The function to integrate is 4*x3 and the interval is [1,2]

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

3. Find the antiderivative of 4*x3

(∫_^)(4*x3*d(x))=4⋅(x4)/4

4. Simplify the antiderivative expression.

4⋅(x4)/4=x4

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

[x4]21=2−1

6. Calculate the final numerical values.

16−1=15

Final Answer

(∫_1^2)(4*x3*d(x))=15

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