Find the Integral (y^5-2y)/(y^3)

Problem

(∫_^)((y^5−2*y)/(y^3)*d(y))

Solution

1. Simplify the integrand by dividing each term in the numerator by the denominator y^3

(y^5)/(y^3)−(2*y)/(y^3)

2. Reduce the powers of y using the laws of exponents.

y^2−2*y^(−2)

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

(∫_^)(y^2*d(y))−(∫_^)(2*y^(−2)*d(y))

4. Use the power rule for integration, (∫_^)(y^n*d(y))=(y^(n+1))/(n+1) for both terms.

(y^3)/3−2*((y^(−1))/(−1))+C

5. Simplify the signs and the expression.

(y^3)/3+2/y+C

Final Answer

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

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