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SolveAlways[(-s^2 + 40 s + 50)/(s (s + 1) (s + 5)^2) == A/s + B/(s + 1) + (C*s + D)/(s + 5)^2, s]

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Apart does what you need to get done but to make this example work you need == between the two expressions and get rid of the last "{s}". Addendum: 1 should be just solve without the Apart. 2 is wrong usage. 3 is incomplete and hence wrong. 4 is correct but it didn't give you what you expected because you setup incorrectly. Since you have $(s+5)^2$ in ...

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Try Apart[]: Apart[(-s^2 + 40 s + 50)/(s (s + 1) (s + 5)^2)] 2/s - 9/(16 (1 + s)) - 35/(4 (5 + s)^2) - 23/(16 (5 + s))

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Note that $$\dfrac{1}{\sec{u}} = \cos{u}$$ so substituting everything gives:  \begin{align*} \int\dfrac{\sqrt{4x^2-1}}{x^2} \, dx &= \int \dfrac{(\tan(u))}{(\sec(x)/2)^2} \cdot \dfrac{1}{2} \tan(u) \sec(u) \, du \\ &= \dfrac{1}{2}\int \dfrac{4 \tan^2(u) \sec(u)}{\sec(u)^2} \,du\\ &= 2\int \dfrac{\tan^2(u)}{\sec(u)} \,du\\ &= 2\int \tan(u) ...

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Most computer programs interpret e^matrix componentwise. In matlab, for example, to do the matrix exponential, you use exp*m*, not exp (which does componentwise exponentiation). This link tells you how to use MatrixExp in Mathematica (and Wolfram Alpha) to do matrix exponentiation.

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The short answer is that WolframAlpha produces different results because Mathematica produces different results: integrand = (x*(x - y)^2)/(x^2 + y^2); anti1 = Integrate[integrand, x, y]; anti2 = Integrate[integrand, y, x]; Simplify[anti1 - anti2] (* Out: 2x^3/9 *) This is not a bug, as the mixed partials of both expressions are the same. ...

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