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
To evaluate this integral, we will use a u-substitution followed by partial fraction decomposition. Step 1: Perform a u-substitution. Let u = e^t. Then, differentiate u with respect to t: du = e^t \, dt Substitute u and du into the integral: (e^t)/((e^t-3)(e^t+3)) \, dt = (1)/((u-3)(u+3)) \, du Step 2: Decompose the integrand using partial fractions. We want to find constants A and B such that: (1)/((u-3)(u+3)) = (A)/(u-3) + (B)/(u+3) Multiply both sides by (u-3)(u+3): 1 = A(u+3) + B(u-3) To find A, set u=3: 1 = A(3+3) + B(3-3) 1 = 6A A = (1)/(6) To find B, set u=-3: 1 = A(-3+3) + B(-3-3) 1 = -6B B = -(1)/(6) So, the partial fraction decomposition is: (1)/((u-3)(u+3)) = (1/6)/(u-3) - (1/6)/(u+3) Step 3: Integrate the partial fractions. ( (1/6)/(u-3) - (1/6)/(u+3) ) \, du = (1)/(6) (1)/(u-3) \, du - (1)/(6) (1)/(u+3) \, du The integral of (1)/(x) is |x|. = (1)/(6) |u-3| - (1)/(6) |u+3| + C Step 4: Combine the logarithmic terms and substitute back u=e^t. Using the logarithm property a - b = ((a)/(b)): = (1)/(6) ( |u-3| - |u+3| ) + C = (1)/(6) |(u-3)/(u+3)| + C Now, substitute back u=e^t: = (1)/(6) |(e^t-3)/(e^t+3)| + C The final answer is: (1)/(6) |(e^t-3)/(e^t+3)| + C Just one more to go for today! Send it over.