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
You're asking a great question about the v \, du part of the integration by parts formula! Let's break down Step 1 for problem 4. The integration by parts formula is u \, dv = uv - v \, du. For problem 4, we set: u = x^2 du = 2x \, dx dv = (2x) \, dx v = (2x) \, dx = (1)/(2)(2x) Now, let's substitute these into the v \, du part of the formula: v \, du = ((1)/(2)(2x)) (2x \, dx) Here, v = (1)/(2)(2x) and du = 2x \, dx. We are integrating their product. To simplify this integral: (1)/(2)(2x) (2x \, dx) = ((1)/(2) · 2) x(2x) \, dx = 1 · x(2x) \, dx = x(2x) \, dx So, the original step was correct: x^2 (2x) dx = (1)/(2)x^2(2x) - x(2x) dx The 2x from du is multiplied with v inside the integral, and then the entire expression is integrated. What's next? Send 'em! 📸