Evaluate the Integral

Problem

(∫_^)((x2+2*x−3)/(x4)*d(x))

Solution

1. Rewrite the integrand by dividing each term in the numerator by the denominator x4

(∫_^)(((x2)/(x4)+(2*x)/(x4)−3/(x4))*d(x))

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

(∫_^)((x(−2)+2*x(−3)−3*x(−4))*d(x))

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

(x(−1))/(−1)+2(x(−2))/(−2)−3(x(−3))/(−3)+C

4. Simplify the coefficients and rewrite the terms with positive exponents.

−x(−1)−x(−2)+x(−3)+C

−1/x−1/(x2)+1/(x3)+C

Final Answer

(∫_^)((x2+2*x−3)/(x4)*d(x))=−1/x−1/(x2)+1/(x3)+C

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