calculus - How does trigonometric substitution work? - Mathematics ...
It isn't "like" u u -substitution: it is u u -substitution, with all the usual u u -substitution trappings such as introducing a new variable that is related to x x by some function. The function just happens to be …
Trigonometic Substitution VS Hyperbolic substitution
Dec 20, 2014 · This doesn't seem to answer OP's question, which asks how to determing a trig or hyperbolic trig substitution is more appropriate (presumably when evaluating integrals), not what …
The intuition behind Trig substitutions in calculus
The key issue is that both the trigonometric and the hyperbolic substitutions lead to simpler integrals, because the irrational integrand becomes a rational function of trigonometric or hyperbolic functions.
Integration of $\\frac{1}{x^2-a^2}$ by trigonometric substitution?
Jul 30, 2018 · Now, I know this can be done by splitting the function into two integrable functions, 1 2a ∫(1 x − a − 1 x + a)dx 1 2 a ∫ (1 x a 1 x + a) d x And then doing the usual stuff. My question is, how …
When to do u-substitution and when to integrate by parts
To reduce powers of trigonometric or exponential functions. Some good examples of the method can be found at Integration by reduction formulae . The general idea is using integral by parts when …
calculus - Making trigonometric substitutions rigorous - Mathematics ...
These comments apply to any trigonometric (or other) substitutions that involve choices or an apparent dependence of the post-substitution calculations on the choices.
Why doesn't trig substitution work for definite integrals?
Sep 27, 2020 · Where have I gone wrong? It seems to me that my logic is faultless, but the definite integral is simply ambiguous when using trig substitutions. This would apply to any integral involving …
Trig substitution; why can we ignore the absolute value?
Jan 25, 2015 · and we substitute x = sin θ x = sin θ then we eventually have to take the square root of cos2 x cos 2 x which is equal to | cos x| | cos x |. But in my textbook and class lectures, we simply …
calculus - Trying to understand why a trigonometric substitution for ...
Mar 7, 2024 · 0 It seems like you're trying to understand why the trigonometric functions sine and cosine are interchangeable in certain contexts. In trigonometry, sine and cosine are related through the unit …
I'm stuck integrating $\int \sqrt {x^2-a^2} dx$ using trigonometric ...
You should use hyperbolic substitution, it is way more convenient than trigonometric here since $\cosh^2-\sinh^2=1$. The inverse hyperbolic functions naturally appear in the final solution.