Evaluate the Integral

Problem

(∫_−2^−1)((u−1)/(u2)*d(u))

Solution

1. Split the integrand by dividing each term in the numerator by the denominator.

(u−1)/(u2)=u/(u2)−1/(u2)

2. Simplify the expression using exponent rules to prepare for integration.

1/u−u(−2)

3. Find the antiderivative using the power rule for u(−2) and the natural logarithm rule for 1/u

(∫_^)((1/u−u(−2))*d(u))=ln(u)−(u(−1))/(−1)

ln(u)+1/u

4. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit −1 and the lower limit −2

(ln(−1)+1/(−1))−(ln(−2)+1/(−2))

5. Simplify the numerical values noting that ln(1)=0

(0−1)−(ln(2)−1/2)

−1−ln(2)+1/2

6. Combine the constants to reach the final result.

−1/2−ln(2)

Final Answer

(∫_−2^−1)((u−1)/(u2)*d(u))=−1/2−ln(2)

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