Integrals of Radical Functions

Trigonometric Substitution

Three mainly cases are listed:

1. When the integrand contains ${\displaystyle \sqrt{a^2-x^2}}$, let ${\displaystyle x=a\sin t}$, ${\displaystyle t}$ in ${\displaystyle [-\frac{\pi }{2},\frac{\pi }{2}]}$, we can get ${\displaystyle \sqrt{a^2-x^2}=a\cos t}$.

   It is sometimes confusing that why we can replace ${\displaystyle x}$ with ${\displaystyle a\sin t}$, which actually derived from the properties of square roots.
   When ${\displaystyle \sqrt{a^2-x^2}}$ exists, we have ${\displaystyle x^2<a^2}$, that is ${\displaystyle -a<x<a}$, so it is sensible to substitute ${\displaystyle x}$ with ${\displaystyle a\sin t}$.

2. When the integrand contains ${\displaystyle \sqrt{x^2+a^2}}$, let ${\displaystyle x=a\tan t}$, ${\displaystyle t}$ in ${\displaystyle [-\frac{\pi }{2},\frac{\pi }{2}]}$, we can get ${\displaystyle \sqrt{x^2+a^2}=a\sec t}$.

3. When the integrand contains ${\displaystyle \sqrt{x^2-a^2}}$, let ${\displaystyle x=a\sec t}$, ${\displaystyle t}$ in ${\displaystyle [0,\frac{\pi }{2})\cup (\frac{\pi }{2},\pi ]}$, we can get ${\displaystyle \sqrt{x^2-a^2}=a\tan t}$.

above are three mainly cases for trigonometric substitution, and they will be useful when handling with cases that contains square roots.

Radical Substitution

For integrands involve ${\displaystyle \sqrt[n]{ax+b}}$, radical substitution can easily simplify the integration.

Let ${\displaystyle u=\sqrt[n]{ax+b}}$, then we have ${\displaystyle u^n=ax+b}$

That is, ${\displaystyle x=\frac{u^n-b}{a}}$, and ${\displaystyle n{u}^{n-1}du=adx}$

So the substitution can be done.

Complete the Square

When the expression ${\displaystyle x^2+Bx+C}$ appears under the radical (expression like ${\displaystyle \sqrt{x^2+Bx+C}}$), a square substitution will prepare it for the trigonometric substitution.

Simply make the quadratic expression into a square with a constant, for example:

${\displaystyle x^2+2x+26=(x+1{)}^2+25}$

Then let ${\displaystyle u=x+1}$, the expression becomes ${\displaystyle u+25}$, then the trigonometric substitution can be applied.

For example, the form ${\displaystyle u^2+C^2}$ is similar with ${\displaystyle {\tan }^2+1={\sec }^2}$, with this identity, we can make the denominator a square of a trig function, instead of polynomials. It makes the integration more easier.