Evaluate the Integral integral of (x^2-4)/x with respect to x

Problem

(∫_^)((x2−4)/x*d(x))

Solution

1. Rewrite the integrand by dividing each term in the numerator by the denominator x

(x2−4)/x=(x2)/x−4/x

2. Simplify the resulting expression to prepare for integration.

(x2)/x−4/x=x−4/x

3. Apply the sum rule for integrals to integrate each term separately.

(∫_^)((x−4/x)*d(x))=(∫_^)(x*d(x))−(∫_^)(4/x*d(x))

4. Integrate using the power rule for x and the logarithmic rule for 1/x

(∫_^)(x*d(x))=(x2)/2

(∫_^)(4/x*d(x))=4*ln(x)

5. Combine the results and add the constant of integration C

(x2)/2−4*ln(x)+C

Final Answer

(∫_^)((x2−4)/x*d(x))=(x2)/2−4*ln(x)+C

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