![Determine the following integrals: 1. integral(1)/(-2x2+20x-47)dx 2. integral3x2xdx 3. integral e^-2x[3]x6dx 4. integral5 4x dx 5. integral^-1((7x)/(3))dx](/_next/image?url=https%3A%2F%2Fohijasnjvprzodcmxltt.supabase.co%2Fstorage%2Fv1%2Fobject%2Fpublic%2Fwhatsapp-images%2F1785262886667-1face4926041facf.png&w=3840&q=75)
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
Step 1: Find the first derivatives of x and y with respect to t. Given x = t^2 and y = 2t^6. (dx)/(dt) = (d)/(dt)(t^2) = 2t (dy)/(dt) = (d)/(dt)(2t^6) = 12t^5 Step 2: Calculate the first derivative (dy)/(dx) using the chain rule. (dy)/(dx) = (dy/dt)/(dx/dt) = (12t^5)/(2t) = 6t^4 Step 3: Calculate the second derivative (d^2y)/(dx^2) using the chain rule. (d^2y)/(dx^2) = (d)/(dx)((dy)/(dx)) = (d)/(dt)((dy)/(dx)) · (dt)/(dx) First, find (d)/(dt)((dy)/(dx)): (d)/(dt)(6t^4) = 24t^3 Next, find (dt)/(dx): (dt)/(dx) = (1)/(dx/dt) = (1)/(2t) Now, multiply these two expressions: (d^2y)/(dx^2) = (24t^3) · ((1)/(2t)) = 12t^2 Step 4: Substitute the given value of t = (1)/(3) into the expression for (d^2y)/(dx^2). (d^2y)/(dx^2) |_t=(1)/(3) = 12((1)/(3))^2 = 12((1)/(9)) = (12)/(9) = (4)/(3) The value of (d^2y)/(dx^2) at t = (1)/(3) is (4)/(3). Send me the next one 📸