Evaluate the Integral

Problem

(∫_0^5)(x*d(x))

Solution

1. Identify the integrand and the limits of integration. The function to integrate is ƒ(x)=x and the interval is [0,5]

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

3. Find the antiderivative of x

(∫_^)(x*d(x))=(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.

[(x2)/2]50

5. Substitute the values x=5 and x=0 into the expression.

5/2−0/2

6. Simplify the resulting numerical expression.

25/2−0=12.5

Final Answer

(∫_0^5)(x*d(x))=25/2

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