Evaluate the Integral

Problem

(∫_^)((3*x2−4)/(x2)*d(x))

Solution

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

(∫_^)(((3*x2)/(x2)−4/(x2))*d(x))

2. Simplify the terms inside the integral.

(∫_^)((3−4*x(−2))*d(x))

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

(∫_^)(3*d(x))−(∫_^)(4*x(−2)*d(x))

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

3*x−4*((x(−1))/(−1))+C

5. Simplify the expression and rewrite the negative exponent as a fraction.

3*x+4/x+C

Final Answer

(∫_^)((3*x2−4)/(x2)*d(x))=3*x+4/x+C

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