Evaluate the Integral

Problem

(∫_−1^1)(x100*d(x))

Solution

1. Identify the integrand and the limits of integration. The function is ƒ(x)=x100 and the interval is [−1,1]

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 x100 by increasing the exponent by 1 and dividing by the new exponent.

(∫_^)(x100*d(x))=(x101)/101

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

[(x101)/101]1(−1)=((1)101)/101−((−1)101)/101

5. Simplify the powers of 1 and −1 Since 101 is an odd number, (−1)101=−1

1/101−(−1)/101

6. Combine the fractions to find the final value.

1/101+1/101=2/101

Final Answer

(∫_−1^1)(x100*d(x))=2/101

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