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
Here are the step-by-step solutions for the first two questions. 1. (16x)/(8x^2 + 2)\,dx Step 1: Use u-substitution. Let u = 8x^2 + 2. Step 2: Differentiate u with respect to x to find du. du = (d)/(dx)(8x^2 + 2)\,dx = (16x)\,dx Step 3: Substitute u and du into the integral. (16x)/(8x^2 + 2)\,dx = (1)/(u)\,du Step 4: Integrate with respect to u. (1)/(u)\,du = |u| + C Step 5: Substitute back u = 8x^2 + 2. |8x^2 + 2| + C Since 8x^2 + 2 is always positive, the absolute value is not strictly necessary. The final answer is (8x^2 + 2) + C. 2. (x^2)/(x^3 + 1)\,dx Step 1: Use u-substitution. Let u = x^3 + 1. Step 2: Differentiate u with respect to x to find du. du = (d)/(dx)(x^3 + 1)\,dx = (3x^2)\,dx Step 3: Rearrange to solve for x^2\,dx. x^2\,dx = (1)/(3)\,du Step 4: Substitute u and x^2\,dx into the integral. (x^2)/(x^3 + 1)\,dx = (1)/(u) · (1)/(3)\,du = (1)/(3) (1)/(u)\,du Step 5: Integrate with respect to u. (1)/(3) (1)/(u)\,du = (1)/(3) |u| + C Step 6: Substitute back u = x^3 + 1. (1)/(3) |x^3 + 1| + C The final answer is (1)/(3) |x^3 + 1| + C.