 About 18,100 results  books.google.com To integrate functions involving square roots, Euler used substitution, although
not our modern trigonometric substitutions. For example, to integrate dx √ α +βx
+γx2 , he considered two cases depending on whether the quadratic polynomial
... 

 books.google.com 1  ablaxbl" Substitute ax = y and this becomes 1  ynb'a dy 1  ybl° ' ~a dx. ...
Recall that just integrating again didn't take Euler any closer to the solution of the
Basel Problem, but, if he divides by x before he integrates, he gets x x2 x3 *•" ! 

 books.google.com –4arctant + C. Table 8.3 Substitutions for integration of irrational functions I
Integral" Substitution n/ Q4: b JR (.. #W da: where r is the lowest common ... Var*
+ ba E c) dr: One of the three Euler substitutions: 1. For a > 0  Var* + by + c = t –
Var 2. 

 books.google.com Euler's Substitutions . . . . . . . . . . . . . . . Other Methods of Integrating Irrational
Expressions Integration of a Binomial Differential Integration of Trigonometric and
Hyperbolic Functions Integration of Certain Irrational Functions with the >u9; ... 

 books.google.com Integration. of. Trigonometric. Functions. No. Integral Method of Integration 10. / R
(x,x n ,...,»• J dx Use the substitution x ... cx+d ~ L 12. f fl(x, \/ax2 + bx + c)dx where
R is a rational function Rationalize it with the Euler substitutions y/ax2 + bx + c ... 

 books.google.com In the case of real zeros integral (8) leads to integration of a rational fraction with
the aid of Euler's third substitution : Wa (a  a 1) (a — a 2) = t (a — a 2), which we
suggest may also be checked by the reader. Euler's substitutions lead to ... 

 books.google.com Instead one has to change the variable of integration, to w say, and then make
the replacement ; so, a correct result of the substitution is ∫ ba(w2 + (x + z)2)dw.
Euler's notion of function of variables, covering both that which is plugged in ... 

 books.google.com [75] Integration of Euler's Dynamical Equations for TorqueFree Motion when A >
B > C Using the integrals of ... One solves these two equations for an and o>3 in
terms of 1o2 and substitutes in Euler's second dynamical equation to get co2 ... 

 books.google.com The integration of the equations, on the supposition that the conditions are
satisfied, may be founded on the preceding investigation (§§ 27—30). One
process follows it directly, and so is a generalisation of Euler's method; another,
after the adoption of a ... Then d6=du+dv—wdyg/dx = 0 , after substitution for olu
and olv from the original equations; and d¢=udv+v0lu—dx =dy, after the same
substitutions. 

 books.google.com ... and second substitutions, and integration by parts ∫ udv=uv− ∫ vdu, which
comes from the product rule for derivatives. ... integrals of rational functions, as
well as standard substitutions such as the trigonometric and Euler's substitutions. 

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