Integrals of the form $ \int\sqrt{\alpha-x^2}dx $ without trigonometric or hyperbolic substitution 4 Doing a standard integral with complex numbers instead of using a trigonometric substitution

2015年10月20日 · $\begingroup$ I suggest that you do not make a trigonometric substitution. If you instead let $$ u=\sqrt{x^2+25} $$ you will end up at $$ \int\frac{1}{u^2-25}\,du=-\frac{1}{5}\,\text{artanh}\,(u/5)+C. $$ $\endgroup$

Good day to everyone, I'm not so good in maths as I wish. But, I'm doing my job to improve it. So, be patience and thanks in advance for any detailed clue. Description of the problem: The arc is

The basic "family" of trigonometric substitutions uses the definitions of trig functions to produce a ratio $ \ \frac{x}{a} = \ $ [trig function] $\theta \ \rightarrow \ x = a \cdot $ [trig function] $\theta \ $ , $ \ a \ $ being a constant.

2016年3月25日 · While I do not know the history on this type of substitution, an early reference were I have seen such a thing used can be found in "A treatise on the integral calculus. Containing the integration of explicit functions of one variable; together with the theory of definite integrals and of elliptic functions " by J. Hymer which was published in ...

There are some basic trigonometric identities which is not hard to memorise, one of the easiest and most important being $\,\,\cos^2x+\sin^2x=1\,\,$ , also known as the Trigonometric Pythagoras Theorem.

Definite Integration with Trigonometric Substitution. 4. Substitution for Trig Integral - GRE Math Subject ...

2015年8月2日 · Trigonometric substitution and triangles. Ask Question Asked 9 years, 6 months ago. Modified 9 years, 6 ...

These comments apply to any trigonometric (or other) substitutions that involve choices or an apparent dependence of the post-substitution calculations on the choices. If all the choices are made explicit and performed consistently there should be no ambiguities.

What's wrong with my approach in evaluating $\int\frac{\mathrm dx}{(a^2x^2-b^2)^\frac32}$ via trigonometric substitution? 1 Reasoning of step in trigonometric integration by substitution question