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
Fresh day m, let's solve. To solve the integral (1-x)/(1+2x) dx using the substitution method: Step 1: Choose a substitution. Let u = 1+2x. Step 2: Find du in terms of dx and express x in terms of u. Differentiating u with respect to x: (du)/(dx) = 2 So, dx = (1)/(2) du. From u = 1+2x, we can express x: u-1 = 2x x = (u-1)/(2) Step 3: Substitute x into the numerator 1-x. 1-x = 1 - ((u-1)/(2)) 1-x = (2 - (u-1))/(2) 1-x = (2 - u + 1)/(2) 1-x = (3-u)/(2) Step 4: Rewrite the integral in terms of u. (1-x)/(1+2x) dx = (3-u)/(2)u ((1)/(2) du) = (3-u)/(4u) du = (1)/(4) ((3)/(u) - (u)/(u)) du = (1)/(4) ((3)/(u) - 1) du Step 5: Integrate with respect to u. = (1)/(4) (3 |u| - u) + C Step 6: Substitute back u = 1+2x. = (1)/(4) (3 |1+2x| - (1+2x)) + C = (3)/(4) |1+2x| - (1)/(4)(1+2x) + C = (3)/(4) |1+2x| - (1)/(4) - (1)/(2)x + C The constant -(1)/(4) can be absorbed into the arbitrary constant C. The final answer is: (3)/(4) |1+2x| - (1)/(2)x + C Send me the next one 📸