Evaluate the Integral

Problem

(∫_1^8)(1/(x3)*d(x))

Solution

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

1/(x3)=x(−3)

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

(∫_^)(x(−3)*d(x))=(x(−3+1))/(−3+1)

(∫_^)(x(−3)*d(x))=(x(−2))/(−2)

3. Simplify the resulting expression back into a fraction form.

(x(−2))/(−2)=−1/(2*x2)

4. Evaluate the definite integral by applying the Fundamental Theorem of Calculus, substituting the upper limit of 8 and the lower limit of 1

[−1/(2*x2)]81=(−1/(2*(8)2))−(−1/(2*(1)2))

5. Calculate the numerical values for each term.

−1/(2*(64))−(−1/2)

−1/128+1/2

6. Find a common denominator to subtract the fractions.

−1/128+64/128

63/128

Final Answer

(∫_1^8)(1/(x3)*d(x))=63/128

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