Evaluate the Integral integral of (x^3-7)/x with respect to x

Problem

(∫_^)((x^3−7)/x*d(x))

Solution

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

(x^3−7)/x=(x^3)/x−7/x

2. Simplify the terms using the properties of exponents.

(x^3)/x−7/x=x^2−7/x

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

(∫_^)((x^2−7/x)*d(x))=(∫_^)(x^2*d(x))−(∫_^)(7/x*d(x))

4. Integrate using the power rule for x^2 and the logarithmic rule for 1/x

(∫_^)(x^2*d(x))=(x^3)/3

(∫_^)(7/x*d(x))=7*ln(x)

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

(x^3)/3−7*ln(x)+C

Final Answer

(∫_^)((x^3−7)/x*d(x))=(x^3)/3−7*ln(x)+C

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