Evaluate the Integral

Problem

(∫_^)((x4−3*x2+5)/(x4)*d(x))

Solution

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

(∫_^)(((x4)/(x4)−(3*x2)/(x4)+5/(x4))*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 (∫_^)(xn*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*x3)+C

Final Answer

(∫_^)((x4−3*x2+5)/(x4)*d(x))=x+3/x−5/(3*x3)+C

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