Evaluate the Integral

Problem

(∫_0^2)(6*x2*d(x))

Solution

1. Identify the constant and the power of x in the integrand 6*x2

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

3. Find the antiderivative by increasing the exponent by 1 and dividing by the new exponent.

(∫_^)(6*x2*d(x))=6⋅(x3)/3

4. Simplify the expression before evaluating the limits.

6⋅(x3)/3=2*x3

5. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit 2 and subtracting the value at the lower limit 0

[2*x3]20=2*(2)3−2*(0)3

6. Calculate the final numerical value.

2*(8)−0=16

Final Answer

(∫_0^2)(6*x2*d(x))=16

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