I need to integrate ∫ x² + 2x + 3/(x-1)(x+2)² dx and I know I need partial fractions first. The general decomposition is: x² + 2x + 3/(x-1)(x+2)² = A/x-1 + B/x+2 + C/(x+2)² But I always make mistakes solving for A, B, and C. Could someone show a clear method — preferably the cover-up method or Heaviside method — for finding these coefficients quickly? What is the general strategy for partial fractions with repeated linear factors?

I'm studying combinatorics and I'm stuck on a problem involving permutations with restrictions. Problem: \"How many ways can 5 distinct mathematics books and 3 distinct physics books be arranged on a shelf if all books of the same subject must be together?\" I know the answer involves 5! \ imes 3! \ imes 2!, but I want to understand why. Also, what if the restriction changes to \"mathematics books must be together\" but physics books can be anywhere? How does the counting change?

I'm studying multivariable calculus and I need to understand the method of Lagrange multipliers for constrained optimization. I understand that to find the extrema of f(x,y) subject to g(x,y) = 0, we solve: \ abla f = λ \ abla g g(x,y) = 0 But why does this work? What is the geometric intuition behind setting the gradients proportional? And how do I apply it to a concrete problem like maximizing f(x,y) = xy subject to x² + y² = 1? Also, how do I determine whether the critical point is a maximum or minimum?

I understand how to compute Laplace transforms using the formula: L f(t) = F(s) = ∫₀^(∞) e^(-st) f(t) dt And I can solve ODEs using Laplace transforms. But I don't have physical intuition for what the Laplace transform actually means. The Fourier transform decomposes a signal into frequencies, which is intuitive. Is there a similar physical interpretation for the Laplace transform? Why do we use s instead of iω?

I need to derive the Maclaurin series for f(x) = e^x from first principles for my AP Calculus BC exam. I know the formula is: f(x) = ∑_(n=0)^(∞) \frac f^((n))(a) n! (x-a)ⁿ For a = 0, this becomes: e^x = ∑_(n=0)^(∞) xⁿ/n! But I want to understand the step-by-step derivation, including why all derivatives of e^x at 0 equal 1, and how to determine the radius of convergence using the ratio test.