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 choose a substitution. The integral to be evaluated is: (6x^2)/(sqrt(1 - (2x^3 + 5)^2)) \, dx This integral resembles the form (du)/(sqrt(1 - u^2)) = (u) + C. Let u = 2x^3 + 5. Step 2: Calculate the differential du. Differentiate u with respect to x: (du)/(dx) = (d)/(dx)(2x^3 + 5) = 6x^2 From this, we can write du = 6x^2 \, dx. Step 3: Substitute u and du into the integral. Substitute u = 2x^3 + 5 and du = 6x^2 \, dx into the original integral: (du)/(sqrt(1 - u^2)) Step 4: Integrate the expression with respect to u. This is a standard integral: (du)/(sqrt(1 - u^2)) = (u) + C Step 5: Substitute back u = 2x^3 + 5 to express the result in terms of x. (2x^3 + 5) + C The phrase "to its 25th term" is not applicable to this type of integration problem, which yields a direct function. We have found the indefinite integral. The final answer is (2x^3 + 5) + C. What's next? Send 'em!