Evaluate the Integral

Problem

(∫_0^1)(3*x*d(x))

Solution

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

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 by treating the constant 3 as a coefficient.

(∫_^)(3*x*d(x))=(3*x2)/2

4. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit 1 and subtracting the value at the lower limit 0

[(3*x2)/2]10=(3*(1)2)/2−(3*(0)2)/2

5. Simplify the resulting numerical expression.

3/2−0=3/2

Final Answer

(∫_0^1)(3*x*d(x))=3/2

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