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 on a roll — To solve the integral (^2 x)/(16 + ^2 x) \, dx, we will use a substitution method. Step 1: Choose a substitution. Let u = x. Step 2: Find the differential du. Differentiate u with respect to x: (du)/(dx) = -^2 x So, du = -^2 x \, dx, which means ^2 x \, dx = -du. Step 3: Substitute u and du into the integral. The integral becomes: (-du)/(16 + u^2) = - (1)/(16 + u^2) \, du Step 4: Evaluate the integral. This integral is in the form (1)/(a^2 + x^2) \, dx = (1)/(a) ((x)/(a)) + C. Here, a^2 = 16, so a = 4. (1)/(4^2 + u^2) \, du = - (1)/(4) ((u)/(4)) + C Step 5: Substitute back u = x. (1)/(4) (( x)/(4)) + C The final answer is -(1)/(4) (( x)/(4)) + C. What's next?