Evaluate the Integral

Problem

(∫_^)((x^4−3*x^2+5)/(x^4)*d(x))

Solution

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

(∫_^)(((x^4)/(x^4)−(3*x^2)/(x^4)+5/(x^4))*d(x))

2. Simplify the expression using the properties of exponents.

(∫_^)((1−3*x^(−2)+5*x^(−4))*d(x))

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

x−(3*x^(−1))/(−1)+(5*x^(−3))/(−3)+C

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

x+3/x−5/(3*x^3)+C

Final Answer

(∫_^)((x^4−3*x^2+5)/(x^4)*d(x))=x+3/x−5/(3*x^3)+C

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