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Note: x > 0 is assumed throughout this article, and the constant of integration is omitted for simplicity. Integrals involving only logarithmic functions
The asymptotic behavior for x → ∞ is = (). where is the big O notation.The full asymptotic expansion is =! ()or / + + () + () +. This gives the following more accurate asymptotic behaviour:
Microsoft is planning to invest about $80 billion in fiscal 2025 on developing data centers to train artificial intelligence models and deploy AI and cloud-based applications, the company said in ...
The natural logarithm can be integrated using integration by parts: = +. Let: u = ln x ⇒ d u = d x x {\displaystyle u=\ln x\Rightarrow du={\frac {dx}{x}}} d v = d x ⇒ v = x {\displaystyle dv=dx\Rightarrow v=x} then: ∫ ln x d x = x ln x − ∫ x x d x = x ln x − ∫ 1 d x = x ln x − x + C {\displaystyle ...
Microsoft acquired the Norwegian enterprise search company Fast Search & Transfer on April 25, 2008, for $1.191 billion (~$1.66 billion in 2023) to boost its search technology. [14] On May 10, 2011, Microsoft announced its acquisition of Skype Technologies, creator of the VoIP service Skype, for $8.5 billion (~$11.4 billion in 2023). [15]
Microsoft is offering several suggestions to the incoming Trump administration and new Congress on handling artificial intelligence (AI) policy in 2025.. In a lengthy blog post published Friday ...
Microsoft invested $1.5 billion in G42 earlier this year, giving the U.S. company a minority stake and a board seat. As part of the deal, G42 would use Microsoft's cloud services to run its AI ...
In integral calculus, integration by reduction formulae is a method relying on recurrence relations. It is used when an expression containing an integer parameter , usually in the form of powers of elementary functions, or products of transcendental functions and polynomials of arbitrary degree , can't be integrated directly.