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
Welcome back lemloyvette458 — been a while! Let's pick up where you left off. To solve the integral (1)/(x^2 + 8x + 25) dx, we will complete the square in the denominator and then use a standard integration formula. Step 1: Complete the square in the denominator. The denominator is x^2 + 8x + 25. To complete the square for x^2 + bx, we add and subtract ((b)/(2))^2. Here, b=8, so ((8)/(2))^2 = 4^2 = 16. x^2 + 8x + 25 = x^2 + 8x + 16 - 16 + 25 x^2 + 8x + 25 = (x+4)^2 + 9 Step 2: Rewrite the integral using the completed square form. (1)/((x+4)^2 + 9) dx Step 3: Apply a substitution to match the standard integral form. Let u = x+4. Then, du = dx. The integral becomes: (1)/(u^2 + 3^2) du This matches the standard integral form (1)/(u^2 + a^2) du = (1)/(a) ((u)/(a)) + C, where a=3. Step 4: Integrate and substitute back. (1)/(3) ((u)/(3)) + C Substitute u = x+4 back into the expression: (1)/(3) ((x+4)/(3)) + C The final answer is (1)/(3) ((x+4)/(3)) + C. Send me the next one 📸