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.
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Here is the solution to the definite integral. We need to evaluate the integral: _0^/A sqrt(1 + (4x)) \, dx Step 1: Apply the trigonometric identity. We use the double angle identity for cosine: 1 + (2) = 2^2(). In our integral, we have 1 + (4x). Let 2 = 4x, which means = 2x. Substituting this into the identity gives: 1 + (4x) = 2^2(2x) Step 2: Substitute the identity into the integral. The integral becomes: _0^/A sqrt(2^2(2x)) \, dx Step 3: Simplify the square root. _0^/A sqrt(2) sqrt(^2(2x)) \, dx = _0^/A sqrt(2) |(2x)| \, dx For this integral to be straightforward, we assume that (2x) 0 over the entire interval of integration [0, ()/(A)]. This typically means that 2x lies in the interval [0, ()/(2)], which implies x lies in [0, ()/(4)]. Therefore, we assume ()/(A) ()/(4), or A 4. Under this assumption, |(2x)| = (2x). So the integral simplifies to: _0^/A sqrt(2) (2x) \, dx Step 4: Integrate the expression. We can pull the constant sqrt(2) out of the integral: sqrt(2) _0^/A (2x) \, dx The integral of (ax) is (1)/(a)(ax). So, the integral of (2x) is (1)/(2)(2x). = sqrt(2) [ (1)/(2)(2x) ]_0^/A Step 5: Evaluate the definite integral using the limits. = sqrt(2)2 [ (2 · ()/(A)) - (2 · 0) ] = sqrt(2)2 [ ((2)/(A)) - (0) ] Since (0) = 0: = sqrt(2)2 [ ((2)/(A)) - 0 ] = sqrt(2)2 ((2)/(A)) The value of the integral is sqrt(2)2 ((2)/(A)). That's 2 down. 3 left today — send the next one.