1
$\begingroup$

My calculus teacher gave us a list of integrals to solve through the substitution method. I've been trying everything for hours, and I just can't find out how to compute these: $∫\frac{x^2}{x^2+4}\,dx\qquad\text{and}\qquad \int\frac{1}{\sqrt{4-x^2}}\,dx.$

Is it even possible to solve them through substitution?

2 Answers 2

7

The first one can be solved by rewriting it first $\int\frac{x^2}{x^2+4}\,dx = \int\frac{x^2+4-4}{x^2+4}\,dx = \int\,dx - 4\int\frac{dx}{x^2+4}.$ Then we can solve the second one of these integrals (first one is easy) by factoring out $4$ from the denominator and then using the substitution $u=\frac{x}{2}$: $\begin{align*} \int\frac{dx}{x^2+4} & = \frac{1}{4}\int\frac{dx}{\left(\frac{x}{2}\right)^2+1}\\ &= \frac{1}{2}\int\frac{du}{u^2+1}.\end{align*}$ The last integral has a direct antiderivative.

For the second integral, try $x=2u$. Then $dx = 2du$, so $\int\frac{1}{\sqrt{4-x^2}}\,dx = \int\frac{2du}{\sqrt{4-4u^2}} = \int\frac{du}{\sqrt{1-u^2}}.$ Does the last one look familiar?

Alternatively, do the same trick: factor out $4$ in the denominator inside the square root: $\begin{align*} \int\frac{1}{\sqrt{4-x^2}}\,dx &= \int\frac{1}{\sqrt{4(1 - (\frac{x}{2})^2}}\,dx\\ &= \int\frac{1}{2\sqrt{1 - (\frac{x}{2})^2}}\end{align*}$ which may suggest the substitution $w=\frac{x}{2}$.


Added. In light of the comments...

My first impulse with the second integral would have been to use a trigonometric substitution. That's because integrals that involve $\sqrt{a^2-x^2}$ but which do not have an $x\,dx$ factor are the traditional proving grounds for the trigonometric substitution. Here, $a=2$, so we would try the substitution $x=2\sin\theta$, with $0\frac{\pi}{2}\leq\theta\leq\frac{\pi}{2}$, $dx = 2\cos\theta\,d\theta$. This wold turn the $\sqrt{4-x^2}$ factor into $\sqrt{4-x^2} = \sqrt{4-4\sin^2\theta} = \sqrt{4(1-\sin^2\theta)} = \sqrt{4\cos^2\theta} = 2|\cos\theta|=2\cos\theta$ with the last equality because $-\frac{\pi}{2}\leq\theta\leq\frac{\pi}{2}$, so $\cos\theta\geq 0$. Then the integral would simplify to $\int\frac{dx}{\sqrt{4-x^2}} = \int\frac{\cos\theta\,d\theta}{2\cos\theta} = \frac{1}{2}\int\,d\theta,$ which is trivial to do; finally, we would return to $x$ using $\theta=\frac{1}{2}\arcsin(x)$.

  • 0
    @Lucas: If that's the case, then the standard way to deal with radicals of the form $\sqrt{a^2-x^2}$ is by the trigonometric substitution $x=a\sin\theta$, which in this instance works (with $a=2$).2012-06-24
0

For the second one, use this substitution if you like:

$x=2 \sin u$. Then $dx=2\cos u\ du$.Substituting that into the second integral,we can reduce the problem like this :

$\int \frac{1}{\sqrt{4-x^2}}dx=\int du=u+C$[where C is the constant of integration]$=\sin^{-1}\frac{x}{2}+C$