Evaluate the Integral

Problem

(∫_1^8)(1/(x2)*d(x))

Solution

1. Rewrite the integrand using a negative exponent to make it easier to apply the power rule for integration.

1/(x2)=x(−2)

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

(∫_^)(x(−2)*d(x))=(x(−2+1))/(−2+1)

3. Simplify the resulting expression before evaluating the limits.

(x(−1))/(−1)=−1/x

4. Evaluate the definite integral by substituting the upper limit of 8 and the lower limit of 1 into the antiderivative.

[−1/x]81=(−1/8)−(−1/1)

5. Calculate the final numerical value by performing the subtraction.

−1/8+1=7/8

Final Answer

(∫_1^8)(1/(x2)*d(x))=7/8

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