Evaluate the Integral

Problem

(∫_^)((x−3)/(x3)*d(x))

Solution

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

(∫_^)((x/(x3)−3/(x3))*d(x))

2. Simplify the terms using the laws of exponents to prepare for integration.

(∫_^)((x(−2)−3*x(−3))*d(x))

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

(x(−1))/(−1)−3*((x(−2))/(−2))+C

4. Simplify the expression by rewriting negative exponents as fractions and combining constants.

−1/x+3/(2*x2)+C

Final Answer

(∫_^)((x−3)/(x3)*d(x))=−1/x+3/(2*x2)+C

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