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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Okay, ThegirlGodishelping🥰💙🫠, let's solve these two problems. 4. Compute _x 0 ( (2x^2 - 3x + 4)/(x) + (5x - 4)/(x) ) Step 1: Combine the fractions since they have a common denominator. _x 0 ( ((2x^2 - 3x + 4) + (5x - 4))/(x) ) Step 2: Simplify the numerator. _x 0 ( (2x^2 - 3x + 5x + 4 - 4)/(x) ) _x 0 ( (2x^2 + 2x)/(x) ) Step 3: Factor out x from the numerator and cancel it with the denominator. Since x 0, but x ≠ 0, we can cancel x. _x 0 ( (x(2x + 2))/(x) ) _x 0 (2x + 2) Step 4: Substitute x=0 into the simplified expression. 2(0) + 2 = 2 The final answer is 2. 5. Integrate and find an exact simplified value for _/6^/3 (1+^2 x)^2 x\,dx Step 1: Simplify the integrand using trigonometric identities. Recall the identity 1+^2 x = ^2 x. So the integrand becomes: ^2 x ^2 x Now, express ^2 x and ^2 x in terms of x and x: (1)/(^2 x) · (1)/(^2 x) = (1)/(^2 x ^2 x) We know that (2x) = 2 x x. Squaring both sides gives ^2(2x) = 4^2 x ^2 x. Therefore, ^2 x ^2 x = (1)/(4)^2(2x). Substitute this back into the integrand: (1)/(1)4^2(2x) = (4)/(^2(2x)) = 4^2(2x) Step 2: Integrate the simplified expression. We need to find 4^2(2x)\,dx. Let u = 2x, so du = 2\,dx, which means dx = (1)/(2)\,du. 4^2(u) ((1)/(2))\,du = 2 ^2(u)\,du The integral of ^2 u is - u. = -2 u + C = -2(2x) + C Step 3: Apply the limits of integration. [ -2(2x) ]_/6^/3 Step 4: Evaluate the expression at the upper and lower limits. At the upper limit x = ()/(3): -2(2 · ()/(3)) = -2((2)/(3)) We know ((2)/(3)) = -(1)/(sqrt(3)). = -2(-(1)/(sqrt(3))) = (2)/(sqrt(3)) At the lower limit x = ()/(6): -2(2 · ()/(6)) = -2(()/(3)) We know (()/(3)) = (1)/(sqrt(3)). = -2((1)/(sqrt(3))) = -(2)/(sqrt(3)) Step 5: Subtract the lower limit value from the upper limit value. (2)/(sqrt(3)) - (-(2)/(sqrt(3))) = (2)/(sqrt(3)) + (2)/(sqrt(3)) = (4)/(sqrt(3)) To rationalize the denominator, multiply by sqrt(3)sqrt(3): (4)/(sqrt(3)) · sqrt(3)sqrt(3) = 4sqrt(3)3 The final answer is 4sqrt(3)3. Last free one today — make it count tomorrow, or type /upgrade for unlimited.