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
I will assume you want me to demonstrate the integration process for Exercise 6 (1)a. This means I will find the function whose indefinite integral is given by the expression in (1)a, and then integrate that function to verify the result. Given answer for Exercise 6 (1)a: F(x) = 4x^3 + 3x^2 - 5x + c. Step 1: Determine the function f(x) that was integrated to obtain F(x). To do this, we find the derivative of F(x). f(x) = (d)/(dx)(4x^3 + 3x^2 - 5x + c) f(x) = 4 · 3x^3-1 + 3 · 2x^2-1 - 5 · 1x^1-1 + 0 f(x) = 12x^2 + 6x - 5 Step 2: Integrate f(x) to verify the given answer. We need to find (12x^2 + 6x - 5)\,dx. Using the power rule for integration, x^n\,dx = x^n+1n+1 + C: (12x^2 + 6x - 5)\,dx = 12x^2\,dx + 6x\,dx - 5\,dx = 12 (x^2+12+1) + 6 (x^1+11+1) - 5x + c = 12 ((x^3)/(3)) + 6 ((x^2)/(2)) - 5x + c = 4x^3 + 3x^2 - 5x + c The result matches the given expression. The verified indefinite integral is: 4x^3 + 3x^2 - 5x + c Send me the next one 📸