Evaluate the Integral integral of (6+x^2)/x with respect to x

Problem

(∫_^)((6+x2)/x*d(x))

Solution

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

(6+x2)/x=6/x+(x2)/x

2. Simplify the resulting terms.

6/x+x

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

(∫_^)(6/x*d(x))+(∫_^)(x*d(x))

4. Apply the power rule and the rule for the natural logarithm to find the antiderivatives.

6*ln(x)+(x2)/2+C

Final Answer

(∫_^)((6+x2)/x*d(x))=6*ln(x)+(x2)/2+C

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