Intereting Posts

Av = Bv for all v implies A = B?
If a Laplacian eigenfunction is zero in an open set, is it identically zero?
Group of order $p^2$ is commutative with prime $p$
List of matrix properties which are preserved after a change of basis
Can exist an even number greater than $36$ with more even divisors than $36$, all of them being a prime$-1$?
Locus using Euclidean geometry
Find the limit $\lim_{n\to\infty} \frac{x_1 y_n + x_2 y_{n-1} + \cdots + x_n y_1}{n}$
How to solve $\int_0^\pi{\frac{\cos{nx}}{5 + 4\cos{x}}}dx$?
How do I determine if a point is within a rhombus?
how many permutations of {1,2,…,9}
Sum up to number $N$ using $1,2$ and $3$
Finding vertices of regular polygon
Calculating $\lim_{n\to\infty}\sqrt{n}\sin(\sin…(\sin(x)..)$
Using Zorn's lemma to show that every field has an algebraic closure.
Updates on Lehmer's Totient Problem

How can I solve this integral: $$\int \frac{1}{x\sqrt{x^2+x}}dx$$ I first completed the square and got: $$\int \frac{1}{x\sqrt{(x+\frac{1}{2})^2-\frac{1}{4}}}dx$$ Then I factored out 1/4 and got: $$2\int \frac{1}{x\sqrt{(2x+1)^2-1}}dx$$ Then I substituted $2x+1$ with $t$ and got: $$2\int \frac{1}{(t-1)\sqrt{t^2-1}}dt$$ I’m not sure what to do next. Please give me a hint ðŸ˜‰

- Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$
- What is $\int_0^1\frac{x^7-1}{\log(x)}\mathrm dx$?
- How to prove for $s<1,|a+b|^s\le|a|^s+|b|^s$
- Maximum and minimum of an integral under integral constraints.
- Correct way to calculate numeric derivative in discrete time?
- Examples of applying L'Hôpitals rule ( correctly ) leading back to the same state?
- The sum of integrals of a function and its inverse: $\int_{0}^{a}f+\int_{0}^{f(a)}f^{-1}=af(a)$
- an indefinite integral $\int \frac{dx}{\sin{x}\sqrt{\sin(2x+\alpha)}}$
- What is a simple example of a limit in the real world?
- Rolle's theorem $\beta \cdot f(x)+f'(x)=0$

Here is an approach.

$$

\begin{align}

\int \frac{1}{x \sqrt{x^2+x}}dx &=\int \frac{1}{\sqrt{1+\frac{1}{x}}}\frac{dx}{x^2}\\\\

&=-\int \frac{1}{\sqrt{1+u}}\:du \quad (u=1/x)\\\\

&=-2\sqrt{u+1}+C\\\\

&=-2\sqrt{\frac{x+1}{x}}+C

\end{align}

$$ where $C$ is any constant.

If you don’t want to have to “spot” which substitution to use, but instead want a “cookbook” approach, then consider using an Euler substitution. This would work here because you have a rational function of $x$ and $\sqrt{ax^2 + bx + c}$, in particular with $a=1$, $b=1$, and $c=0$.

You can use Euler’s first substitution if $a>0$: here we write $\sqrt{ax^2+bx+c} = \pm x\sqrt{a}+t$ which gives $x = \frac{c-t^2}{\pm 2t\sqrt{a}-b}$. Since $a=1$ that’s an option, but neither an $x+t$ nor $-x+t$ in the denominator look particularly helpful here because of the multiplication by $x$. If that had been an addition or subtraction of $x$ instead, then this substitution would have been ideal as that would have cancelled.

According to Wikipedia, we can use Euler’s second substitution if $c>0$: here we write $\sqrt{ax^2+bx+c} = xt\pm\sqrt{c}$ which gives $x = \frac{\pm 2t\sqrt{c}-b}{a-t^2}$. Since $c=0$ I will not pursue this.

Euler’s third substitution is used if $ax^2+bx+c$ has real roots $\alpha$ and $\beta$; we write $\sqrt{ax^2 + bx + c} = \sqrt{a(x-\alpha)(x-\beta)} = (x-\alpha)t$ which gives $x = \frac{a\beta-\alpha t^2}{a-t^2}$. This multiplicative form seems more suited for our purposes. In our case $x^2 + x = x(x+1)$ so we can take $\alpha=0$ and $\beta=-1$.

Pursuing the third substitution we obtain immediately $\sqrt{x^2 + x} = \sqrt{x(x+1)} = xt$ whence $t=\frac{\sqrt{x(x+1)}}{x} = \sqrt{\frac{x+1}{x}}$ and $x=\frac{1(-1) – 0t^2}{1-t^2}=(t^2-1)^{-1}$. Then $\frac{\mathrm{d}x}{\mathrm{d}t}=-2t(t^2-1)^{-2}$. Putting this together,

$$\int \frac{\mathrm{d}x}{x\sqrt{x(x+1)}}= \int \frac{-2t(t^2-1)^{-2}\mathrm{d}t}{x^2t} = \int \frac{-2t(t^2-1)^{-2}\mathrm{d}t}{(t^2-1)^{-2} t} = \int -2 \mathrm{d}t = -2t + C$$

Expressing back in terms of $x$ gives $-2 \sqrt{\frac{x+1}{x}} + C$. The advantage of this approach is that no “cleverness” is needed to find the required substitution, we just work through a checklist and see which one applies. A cleverer solution might be faster to perform, but take longer to spot.

Going back to Euler’s second substitution, we can see that even though $c=0$ this would have worked just fine, and in fact it would have immediately given the same result as Euler’s third substitution. I have the feeling the “$>$” in the Wikipedia article should really say “$\geq$” – but it doesn’t make difference, since when $c=0$ the quadratic would factorise with one of the roots as zero and we can apply the third substitution as above.

Finally, reconsidering the first substitution, we could have gone ahead and put $\sqrt{x^2 + x} = x + t$ and $x = \frac{-t^2}{2t-1} = t^2 (1-2t)^{-1}$. By product rule,

$$\frac{\mathrm{d}x}{\mathrm{d}t}=2t(1-2t)^{-1} + 2t^2(1-2t)^{-2} = 2t(1-t)(1-2t)^{-2}$$

Substituting into the integral and clearing up is rather messy, for the reasons mentioned above, but can be done. Skipping quite a lot of algebra,

$$\int \frac{\mathrm{d}x}{x\sqrt{x(x+1)}}= \int \frac{2t(1-t)(1-2t)^{-2}\mathrm{d}t}{t^2 (1-2t)^{-1} ( t^2 (1-2t)^{-1} + t)} = \int 2t^{-2} \mathrm{d}t = -2t^{-1} + C$$

In terms of $x$, this is $\frac{2}{x – \sqrt{x^2 + x}} + C$. This is an alternative form to that given by Euler’s third substitution, but was harder work to acquire. So don’t just jump straight for the first substitution you see that fulfils the necessary conditions on the coefficients; spend a moment considering which of the available options best fits the form of the question.

To do integrations of the form $\int \frac {dx}{P(x)\sqrt{ Q(x)}}$ where P(x) is linear and Q(x) is quadratic, you can substitute $ P(x) = u $. Substituting $t-1 = \frac{1}{u}$ reduces the integral to $$\frac{1}{2}\int \frac {-du}{\sqrt{1 +2u}}$$

Now this can be integrated further.

Take x^2 our from that under the square root sign and it will transform into `integral dx/{x^2 root`

(1+1/x)} . N ow take – common and take 1+1/x as t^2 then it becomes a trivial tast to perform the integration

- Closed form for $n$th derivative of exponential of $f$
- Wiggly polynomials
- When does $\sqrt{wz}=\sqrt{w}\sqrt{z}$?
- Are there $n$ groups of order $n$ for some $n>1$?
- Matlab's “buggy” symbolic integration function seems to be equivalent to the Implied Integral. Coincedence or intentional?
- Questions about geometric distribution
- $\cap_{A \in \mathcal{F}}(B \cup A) \subseteq B \cup (\cap \mathcal{F})$
- Triangulations of n-gon
- Using the Reflection Theorem
- Is every non-square integer a primitive root modulo some odd prime?
- Counting integer partitions of n into exactly k distinct parts size at most M
- Bounded index of nilpotency
- a conjecture of certain q-continued fractions
- Independence of a random variable $X$ from itself
- Solving an inequality : $n \geq 3$ , $n^{n} \lt (n!)^{2}$.