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: Identify the integral and simplify the integrand. The integral to evaluate is: _0^ e^x - e^-xe^x + e^-x \,dx The integrand can be recognized as the hyperbolic tangent function, x. Recall that x = e^x - e^-x2 and x = e^x + e^-x2. Therefore, e^x - e^-xe^x + e^-x = (2 x)/(2 x) = x. The integral becomes: _0^ x \,dx Step 2: Find the indefinite integral of x. We know that x = ( x)/( x). Let u = x. Then, du = x \,dx. Substituting these into the integral: (1)/(u) \,du = |u| + C Since x = e^x + e^-x2 is always positive for real x, we can write: x \,dx = ( x) + C Step 3: Evaluate the improper integral using limits. Since the upper limit is , this is an improper integral. We evaluate it using a limit: _0^ x \,dx = _b _0^b x \,dx Substitute the indefinite integral: = _b [ ( x) ]_0^b Apply the limits of integration: = _b (( b) - ( 0)) Step 4: Calculate the values at the limits. First, evaluate 0: 0 = e^0 + e^-02 = (1+1)/(2) = 1 So, ( 0) = (1) = 0. Next, evaluate the limit of ( b) as b : As b , e^b and e^-b 0. Therefore, b = e^b + e^-b2 as b . Since (y) as y , we have: _b ( b) = Step 5: Conclude the result. Substitute these values back into the expression from Step 3: _0^ x \,dx = - 0 = Since the limit is infinite, the integral diverges. The final answer is Diverges.