Trig substitutions. What is going on?

Question

I am a bit confused about my book's explanation regarding trig substitutions. I understand u-subs... but I don't get this:

first where did the equation $x = a \sin\theta$ come from? What is x here? What does a represent?

Answer 1

You are trying to compute $$\int \sqrt{1-x^2}dx.$$ Notice that if $x$ were nice and looked like $\sin t$, then the root would conveniently disappear since $$\sqrt{1-x^2} = \sqrt{1 - \sin^2 t} = \sqrt{\cos^2 t} = \cos t,$$ so substitution $x = \sin t$ makes such an integral much easier to handle.

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In a more general case, where you want to find $$\int \sqrt{a^2-x^2}dx$$ for some fixed positive real number $a$, the trick is to represent $x$ in such a form as to factor $a$ outside of the root, allowing us to do a similar trick as before. So we let $x = a\sin t$, to get $$ \sqrt{a^2-x^2} = \sqrt{a^2 - (a\sin t)^2} = \sqrt{a^2 - a^2 \sin^2 t} = a \sqrt{1 - \sin^2 t} = a \cos t. $$

Answer 2

The theorem $$\int_{g(a)}^{g(b)} f(x)dx=\int_a^b f(g(x)) g'(x) dx$$ (with the appropriate hypotheses) has various names. When used in the specific direction of right-to-left in one dimension, it's usually called $u$-substitution (and one usually uses $u$ instead of $x$ as the variable of integration).

However, the other direction, left-to-right, is just as useful. For special instances of $g$, this result is called various things. If $g$ is a trigonometric function, such as $\sin$, $\cos$, or $\tan$, this may be called trigonometric substitution. If $h$ is a hyperbolic function, such as $\sinh$, $\cosh$, or $\tanh$, it may be called hyperbolic substitution. On the whole however, the left-to-right direction (or both directions together) is often called change of variables---especially in higher dimensions.