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
You're on a roll — Step 1: Rewrite the integrand using negative exponents. The integral is _0^1 (4)/(x^2) dx. We can rewrite (4)/(x^2) as 4x^-2. So the integral becomes: _0^1 4x^-2 dx Step 2: Recognize this as an improper integral. The integrand f(x) = (4)/(x^2) is undefined at x=0, which is one of the limits of integration. Therefore, this is an improper integral of Type 2. We evaluate it using a limit: _0^1 4x^-2 dx = _a 0^+ _a^1 4x^-2 dx Step 3: Find the antiderivative of 4x^-2. Using the power rule for integration, x^n dx = x^n+1n+1 + C: 4x^-2 dx = 4 ( x^-2+1-2+1 ) = 4 ( x^-1-1 ) = -4x^-1 = -(4)/(x) Step 4: Evaluate the definite integral from a to 1. [ -(4)/(x) ]_a^1 = ( -(4)/(1) ) - ( -(4)/(a) ) = -4 + (4)/(a) Step 5: Take the limit as a 0^+. _a 0^+ ( -4 + (4)/(a) ) As a approaches 0 from the positive side, (4)/(a) approaches positive infinity. _a 0^+ ( -4 + (4)/(a) ) = -4 + = Since the limit is , the integral diverges. The final answer is Diverges. Just one more to go today! Send it over.