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: Express the fraction in partial fractions. Let (7-x)/((1+x)(2+x)) = (A)/(1+x) + (B)/(2+x). Multiply both sides by (1+x)(2+x): 7-x = A(2+x) + B(1+x) To find A, set x = -1: 7 - (-1) = A(2 - 1) + B(1 - 1) 8 = A(1) + B(0) A = 8 To find B, set x = -2: 7 - (-2) = A(2 - 2) + B(1 - 2) 9 = A(0) + B(-1) 9 = -B B = -9 So, the partial fraction decomposition is: (7-x)/((1+x)(2+x)) = (8)/(1+x) - (9)/(2+x) Step 2: Integrate the partial fractions. We need to evaluate (7-x)/((1+x)(2+x)) dx. Using the partial fractions from Step 1: ( (8)/(1+x) - (9)/(2+x) ) dx Integrate each term: (8)/(1+x) dx = 8 |1+x| (9)/(2+x) dx = 9 |2+x| Combine the results: (7-x)/((1+x)(2+x)) dx = 8 |1+x| - 9 |2+x| + C Using logarithm properties (a b = b^a and a - b = (a)/(b)): 8 |1+x| - 9 |2+x| + C = |(1+x)^8| - |(2+x)^9| + C = | ((1+x)^8)/((2+x)^9) | + C The question asks to show that the integral is $ | (1+x)^9