Тригонометричні заміни

In the last section, we investigated how a change of variables can turn a complicated integral into a simple one. We did this by setting a part of the integrand equal to $u$, a variable in terms of $x$. However, if we instead write $x$ as a function of $u$, often a trigonometric one, we can turn a seemingly unsolvable integral into a more manageable one. The problem with these substitutions, however, is that the best substitution can be hard to guess!

Take the integral $\int\frac{dx}{1+x^2}$ for example. Without substitution, it looks impossible to solve! Now the most useful substitution here will seem like a complete non sequitur, but see if you can figure it out before I show you!

The best substitution to use here is $x = \tan\theta$. In this case, $dx = \sec^2\theta d\theta$. The integral becomes

$\int \frac{dx}{1+x^2} = \int \frac{\sec^2 \theta}{1+\tan^2 \theta}d\theta = \int d\theta = \theta = \tan^{-1} x$.

Let's try another one.

Choosing $x = a \sin \theta$ and $dx = a\cos\theta d\theta$ simplifies the integral to

$\int a^2 \cos^2 \theta d\theta = \frac{a^2}{2}\theta + \frac{1}{2}a\sin\theta(a\cos\theta) = \frac{a^2}{2}\sin^{-1}x + \frac{1}{2}x\sqrt{a^2 - x^2}$.

As the above problem demonstrates, sometimes extensive manipulation of trigonometric identities is needed to get the integral back in terms of $x$.