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.
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Hey sthamntungwa — ready when you are. Here are the solutions for the integration problems: 16. To evaluate the integral x(x + (1)/(x))\,dx: Step 1: Distribute x into the parenthesis. x(x + (1)/(x)) = x · x + x · (1)/(x) = x^2 + 1 Step 2: Integrate the simplified expression term by term. (x^2 + 1)\,dx = x^2\,dx + 1\,dx Step 3: Apply the power rule for integration x^n\,dx = x^n+1n+1 + C. x^2\,dx = x^2+12+1 + C_1 = (x^3)/(3) + C_1 1\,dx = x + C_2 Step 4: Combine the results and add the constant of integration C. x(x + (1)/(x))\,dx = (x^3)/(3) + x + C The integral is (x^3)/(3) + x + C. 17. To evaluate the integral (x^2 + 2x + 1)/(x+1)\,dx: Step 1: Factor the numerator. The numerator x^2 + 2x + 1 is a perfect square trinomial, which can be factored as (x+1)^2. (x^2 + 2x + 1)/(x+1) = ((x+1)^2)/(x+1) Step 2: Simplify the expression by canceling out one (x+1) term, assuming x ≠ -1. ((x+1)^2)/(x+1) = x+1 Step 3: Integrate the simplified expression term by term. (x+1)\,dx = x\,dx + 1\,dx Step 4: Apply the power rule for integration. x\,dx = x^1+11+1 + C_1 = (x^2)/(2) + C_1 1\,dx = x + C_2 Step 5: Combine the results and add the constant of integration C. (x^2 + 2x + 1)/(x+1)\,dx = (x^2)/(2) + x + C The integral is (x^2)/(2) + x + C. What's next?