# Primitive function - smart substitution

• Rectifier
In summary: So it seems like the best substitution to make here is to use the first one.In summary, the problem is to find an equation for the integral ##\int \frac{x}{\sqrt{x^2+2x+10}}##. The attempt using substitution ##\tan(u) = \frac {x + 1} 3## resulted in an easier problem to solve.
Rectifier
Gold Member
The problem
$$\int \frac{x}{\sqrt{x^2+2x+10}} \ dx$$

The attempt

## \int \frac{x}{\sqrt{x^2+2x+10}} \ dx = \int \frac{x}{\sqrt{(x+1)^2+9}} \ dx##

Is there any smart substitution I can make here to make this a bit easier to solve?

Rectifier said:
The problem
$$\int \frac{x}{\sqrt{x^2+2x+10}} \ dx$$

The attempt

## \int \frac{x}{\sqrt{x^2+2x+10}} \ dx = \int \frac{x}{\sqrt{(x+1)^2+9}} \ dx##

Is there any good substitution I can make here to make this a bit easier to solve?
Looks like this trig substitution might work, with ##\tan(u) = \frac {x + 1} 3##

Rectifier
Oh, okay so I basically get ##\int 3 \tan u - 1 \ du## after that and that is much easier. Very elegant substitution indeed.

The list of formulas suggests (by looking at the result) to first do an integration by parts to get rid of the ##x## in the nominator, and then some logarithm with ##z^2:=x^2+2x+10##. (But I only took a brief look.)

If we try to look for alternative substitutions;

That integral from my problem looks a lot like an standard-integral which I have in my book:

## \int \frac{1}{\sqrt{x^2+a}} = \ln | x + \sqrt{x^2+a} | ## but the only thing different is the x in the nominator so I guess I could do this by integrating by parts and removing the x as a derivative inside the integral but I will end up with a part with ##...-\int {\ln | x + \sqrt{x^2+a}}| \ dx## which is not much easier to solve

Rectifier said:
The problem
$$\int \frac{x}{\sqrt{x^2+2x+10}} \ dx$$

The attempt

## \int \frac{x}{\sqrt{x^2+2x+10}} \ dx = \int \frac{x}{\sqrt{(x+1)^2+9}} \ dx##

Is there any smart substitution I can make here to make this a bit easier to solve?
Writing the numerator as ##x = \frac{1}{2} (2x +2) -1## we have
$$\int \frac{x}{\sqrt{(x+1)^2+9}} \, dx = \frac{1}{2} \int \frac{d(x+1)^2}{\sqrt{(x+1)^2+9}} - \int \frac{dx}{\sqrt{(x+1)^2+9}}$$
The first one has the form ##(1/2) \int du/\sqrt{u+9}##, while the second one has the form ##\int du/\sqrt{u^2+9}##.

lurflurf and Rectifier

## 1. What is a primitive function?

A primitive function is a basic mathematical function that cannot be further simplified or broken down into simpler functions. Examples of primitive functions include addition, subtraction, multiplication, and division.

## 2. What is smart substitution in relation to primitive functions?

Smart substitution is a technique used in calculus to simplify complex mathematical expressions involving primitive functions. It involves substituting a new variable, usually denoted by "u", for a portion of the original expression in order to make it easier to integrate.

## 3. How does smart substitution work?

Smart substitution works by choosing a new variable that will help to simplify the original expression. This new variable is usually chosen based on its ability to cancel out or reduce the complexity of the expression. After the substitution is made, the integral can be solved using standard integration techniques.

## 4. What are the benefits of using smart substitution?

Using smart substitution can make it easier to solve complex integrals involving primitive functions. It can also help to avoid mistakes and reduce the amount of time needed to solve the integral. Additionally, it can help to reveal patterns in the expression that can lead to a more efficient solution.

## 5. Are there any limitations to smart substitution?

While smart substitution can be a useful tool in solving integrals, it is not a foolproof method. It may not always work for every expression, and in some cases, it may actually make the integral more complicated. It is important to use discretion and consider other integration techniques when using smart substitution.

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