This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Simplify the integrand. First, factor out y from the numerator and denominator: (y^3+4y^2)/(y^3+y) = (y^2(y+4))/(y(y^2+1)) = (y(y+4))/(y^2+1) = (y^2+4y)/(y^2+1) Since the degree of the numerator is equal to the degree of the denominator, perform polynomial long division or rewrite the numerator: (y^2+4y)/(y^2+1) = ((y^2+1) + 4y - 1)/(y^2+1) = 1 + (4y-1)/(y^2+1) So the integral becomes: ( 1 + (4y-1)/(y^2+1) ) dy Step 2: Split the integral into simpler parts. ( 1 + (4y-1)/(y^2+1) ) dy = 1 \, dy + (4y)/(y^2+1) \, dy - (1)/(y^2+1) \, dy Step 3: Evaluate each integral. The first integral is straightforward: 1 \, dy = y For the second integral, (4y)/(y^2+1) \, dy, use a substitution. Let u = y^2+1, so du = 2y \, dy. Then 4y \, dy = 2 \, du. (4y)/(y^2+1) \, dy = (2 \, du)/(u) = 2 |u| + C_1 = 2 (y^2+1) + C_1 (Since y^2+1 is always positive, we can remove the absolute value.) The third integral is a standard arctangent form: (1)/(y^2+1) \, dy = (y) + C_2 Step 4: Combine the results. (y^3+4y^2)/(y^3+y) dy = y + 2 (y^2+1) - (y) + C The final answer is y + 2 (y^2+1) - (y) + C. Got more? Send 'em!