Evaluate the Integral

Problem

(∫_^)((x^2+2*x−3)/(x^4)*d(x))

Solution

1. Rewrite the integrand by dividing each term in the numerator by the denominator x^4

(∫_^)(((x^2)/(x^4)+(2*x)/(x^4)−3/(x^4))*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 (∫_^)(x^n*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/(x^2)+1/(x^3)+C

Final Answer

(∫_^)((x^2+2*x−3)/(x^4)*d(x))=−1/x−1/(x^2)+1/(x^3)+C

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