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 are the solutions for the problems: Problem 50: We need to evaluate the integral ((5x))/(sqrt(1-5x^2)) dx. Assumption: Based on the typical form for integrals involving , we assume the denominator is sqrt(1-(5x)^2) = sqrt(1-25x^2). If the denominator is strictly sqrt(1-5x^2), the integral would be significantly more complex and not a direct u-substitution. Step 1: Use u-substitution. Let u = (5x). Step 2: Find the differential du. Recall that (d)/(dx)((ax)) = (a)/(sqrt(1-(ax)^2)). So, differentiating u with respect to x gives: du = (5)/(sqrt(1-(5x)^2)) dx = (5)/(sqrt(1-25x^2)) dx Rearrange to solve for (1)/(sqrt(1-25x^2)) dx: (1)/(5) du = (1)/(sqrt(1-25x^2)) dx Step 3: Substitute u and du into the integral. The integral becomes: u · (1)/(5) du = (1)/(5) u \, du Step 4: Integrate with respect to u. (1)/(5) · (u^2)/(2) + C = (u^2)/(10) + C Step 5: Substitute back u = (5x). (((5x))^2)/(10) + C The final answer is (((5x))^2)/(10) + C. Problem 51: We need to find the exact value of sech( 2). Step 1: Recall the definition of sech(x). sech(x) = (1)/((x)) And the definition of (x): (x) = e^x + e^-x2 Step 2: Substitute x = 2 into the definition of (x). ( 2) = e^ 2 + e^- 22 Step 3: Simplify the exponential terms. Using the property e^ a = a: e^ 2 = 2 Using the property e^-a = (1)/(e^a): e^- 2 = e^(2^-1) = 2^-1 = (1)/(2) Step 4: Substitute these values back into the expression for ( 2). ( 2) = (2 + 1)/(2)2 Step 5: Calculate the value of ( 2). ( 2) = (4)/(2) + (1)/(2)2 = (5)/(2)2 = (5)/(4) Step 6: Calculate sech( 2). sech( 2) = (1)/(( 2)) = (1)/(5)4 = (4)/(5) The final answer is (4)/(5). That's 2 down. 3 left today — send the next one.