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
The power rule for integration states that for any real number n ≠ -1: x^n dx = x^n+1n+1 + C where C is the constant of integration. To show this rule, we can differentiate the result x^n+1n+1 + C with respect to x and demonstrate that it returns the original function x^n. Step 1: Start with the proposed integral result. Let F(x) = x^n+1n+1 + C. Step 2: Differentiate F(x) with respect to x. (d)/(dx) ( x^n+1n+1 + C ) Step 3: Apply the constant multiple rule and the power rule for differentiation, (d)/(dx)(x^k) = kx^k-1. The derivative of a constant C is 0. (d)/(dx) ( (1)/(n+1) x^n+1 + C ) = (1)/(n+1) · (d)/(dx)(x^n+1) + (d)/(dx)(C) = (1)/(n+1) · (n+1)x^(n+1)-1 + 0 Step 4: Simplify the expression. = (1)/(n+1) · (n+1)x^n = x^n Since the derivative of x^n+1n+1 + C is x^n, the power rule for integration is shown to be correct. Last free one today — make it count tomorrow, or type /upgrade for unlimited.