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
Hey TALA, good to see you again. Here are the solutions for the integrals: Problem 46: We need to evaluate the integral ( x)/(3 + x) dx. Step 1: Use u-substitution. Let u = 3 + x. Step 2: Find the differential du. Differentiating u with respect to x gives: du = x \, dx Step 3: Substitute u and du into the integral. The integral becomes: (1)/(u) du Step 4: Integrate with respect to u. The integral of (1)/(u) is |u|. (1)/(u) du = |u| + C Step 5: Substitute back u = 3 + x. |3 + x| + C The final answer is |3 + x| + C. Problem 47: We need to evaluate the integral (^2(2x))/(1 + (2x)) dx. Step 1: Use u-substitution. Let u = 1 + (2x). Step 2: Find the differential du. Differentiating u with respect to x using the chain rule gives: du = ^2(2x) · (d)/(dx)(2x) \, dx du = ^2(2x) · 2 \, dx Rearrange to solve for ^2(2x) \, dx: (1)/(2) du = ^2(2x) \, dx Step 3: Substitute u and du into the integral. The integral becomes: (1)/(u) · (1)/(2) du (1)/(2) (1)/(u) du Step 4: Integrate with respect to u. (1)/(2) |u| + C Step 5: Substitute back u = 1 + (2x). (1)/(2) |1 + (2x)| + C The final answer is (1)/(2) |1 + (2x)| + C. Send me the next one 📸