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 Fawwaz — been a while! Let's pick up where you left off. Here are the solutions to the exercises: 1) Evaluate 2x \, dx. Step 1: Let u = 2x. Step 2: Differentiate u with respect to x to find du. (du)/(dx) = 2 dx = (1)/(2) du Step 3: Substitute u and dx into the integral. u · (1)/(2) du = (1)/(2) u \, du Step 4: Integrate with respect to u. (1)/(2) u + C Step 5: Substitute back u = 2x. (1)/(2) (2x) + C 2) Evaluate e^2x-4 \, dx. Step 1: Let u = 2x-4. Step 2: Differentiate u with respect to x to find du. (du)/(dx) = 2 dx = (1)/(2) du Step 3: Substitute u and dx into the integral. e^u · (1)/(2) du = (1)/(2) e^u \, du Step 4: Integrate with respect to u. (1)/(2) e^u + C Step 5: Substitute back u = 2x-4. (1)/(2) e^2x-4 + C 3) Evaluate e^3x \, dx. Step 1: Let u = 3x. Step 2: Differentiate u with respect to x to find du. (du)/(dx) = 3 dx = (1)/(3) du Step 3: Substitute u and dx into the integral. e^u · (1)/(3) du = (1)/(3) e^u \, du Step 4: Integrate with respect to u. (1)/(3) e^u + C Step 5: Substitute back u = 3x. (1)/(3) e^3x + C 4) Evaluate (2x+3)^1/2 \, dx. Step 1: Let u = 2x+3. Step 2: Differentiate u with respect to x to find du. (du)/(dx) = 2 dx = (1)/(2) du Step 3: Substitute u and dx into the integral. u^1/2 · (1)/(2) du = (1)/(2) u^1/2 \, du Step 4: Integrate with respect to u using the power rule x^n \, dx = x^n+1n+1 + C. (1)/(2) · u^1/2+11/2+1 + C = (1)/(2) · u^3/23/2 + C Step 5: Simplify the expression. (1)/(2) · (2)/(3) u^3/2 + C = (1)/(3) u^3/2 + C Step 6: Substitute back u = 2x+3. (1)/(3) (2x+3)^3/2 + C Send me the next one 📸