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 to your problems. Problem 1: Differentiation Given y = (x^2+1)sqrt( x)e^3x, find (dy)/(dx). Step 1: Take the natural logarithm of both sides and simplify using logarithm properties. y = ( (x^2+1)sqrt( x)e^3x ) y = (x^2+1) + (sqrt( x)) - (e^3x) y = (x^2+1) + (1)/(2)( x) - 3x Step 2: Differentiate both sides with respect to x. (1)/(y)(dy)/(dx) = (d)/(dx)((x^2+1)) + (d)/(dx)((1)/(2)( x)) - (d)/(dx)(3x) (1)/(y)(dy)/(dx) = (2x)/(x^2+1) + (1)/(2) · (1)/( x) · (1)/(x) - 3 (1)/(y)(dy)/(dx) = (2x)/(x^2+1) + (1)/(2x x) - 3 Step 3: Solve for (dy)/(dx) by multiplying by y and substituting the original expression for y. (dy)/(dx) = y ( (2x)/(x^2+1) + (1)/(2x x) - 3 ) (dy)/(dx) = (x^2+1)sqrt( x)e^3x ( (2x)/(x^2+1) + (1)/(2x x) - 3 ) The final answer is (dy)/(dx) = (x^2+1)sqrt( x)e^3x ( (2x)/(x^2+1) + (1)/(2x x) - 3 ). Problem 2: Integration Evaluate I = (x^3 x)/((1+x^4)^2) dx. Step 1: Use integration by parts, u \, dv = uv - v \, du. Let u = x and dv = (x^3)/((1+x^4)^2) dx. Step 2: Calculate du and v. du = (1)/(x) dx To find v, integrate dv: v = (x^3)/((1+x^4)^2) dx Let w = 1+x^4, so dw = 4x^3 dx x^3 dx = (1)/(4) dw. v = (1)/(w^2) · (1)/(4) dw = (1)/(4) w^-2 dw = (1)/(4) ( w^-1-1 ) = -(1)/(4w) Substitute w = 1+x^4 back: v = -(1)/(4(1+x^4)) Step 3: Apply the integration by parts formula. I = ( x) (-(1)/(4(1+x^4))) - (-(1)/(4(1+x^4))) ((1)/(x) dx) I = -( x)/(4(1+x^4)) + (1)/(4) (1)/(x(1+x^4)) dx Step 4: Evaluate the remaining integral (1)/(x(1+x^4)) dx. Multiply the numerator and denominator by x^3: (x^3)/(x^4(1+x^4)) dx Let t = x^4, so dt = 4x^3 dx x^3 dx = (1)/(4) dt. The integral becomes: (1)/(4) (1)/(t(1+t)) dt Use partial fractions for (1)/(t(1+t)): (1)/(t(1+t)) = (A)/(t) + (B)/(1+t) 1 = A(1+t) + Bt Setting t=0 A=1. Setting t=-1 B=-1. So, (1)/(t(1+t)) = (1)/(t) - (1)/(1+t). (1)/(4) ( (1)/(t) - (1)/(1+t) ) dt = (1)/(4) (|t| - |1+t|) + C' = (1)/(4) |(t)/(1+t)| + C' Substitute t=x^4 back (assuming x>0): (1)/(4) ((x^4)/(1+x^4)) + C' Step 5: Combine the results to get the final integral. I = -( x)/(4(1+x^4)) + (1)/(4) ( (1)/(4) ((x^4)/(1+x^4)) ) + C I = -( x)/(4(1+x^4)) + (1)/(16) ((x^4)/(1+x^4)) + C The final answer is I = -( x)/(4(1+x^4)) + (1)/(16) ((x^4)/(1+x^4)) + C. Send me the next one 📸