Understanding Integration of 1/(x^2 + a^2) and the Role of the Tan Function

[lioric]

I cannot understand the intergration done here
The part how 1/a came, what happened to the x and how did tan come into this

 

[Samy_A]

View attachment 96316
I cannot understand the intergration done here
The part how 1/a came, what happened to the x and how did tan come into this

The x went away because it is the dummy integration variable in a definite integral.

For starters: do you know how to evaluate the following indefinite integral: $\int \frac{1}{1+x²}dx$?

 

[lioric]

No

The x went away because it is the dummy integration variable in a definite integral.

For starters: do you know how to evaluate the following indefinite integral: $\int \frac{1}{1+x²}dx$?

 

[Samy_A]

No

Do you know how to use a substitution in order to compute an integral?

 

[lioric]

Yes

Do you know how to use a substitution in order to compute an integral?

 

[Samy_A]

Yes

Fine. So start with the indefinite integral $\int \frac{1}{1+x²}dx$ and use the substitution $x=\tan y$ to compute it.

 

[lioric]

Fine. So start with the indefinite integral $\int \frac{1}{1+x²}dx$ and use the substitution $x=\tan y$ to compute it.

Thank you very much

 

[lioric]

This was as far as I could go
I'm wondering how that 1/a came and how to make this into a 1/x^2+1 formate so I can input tan
Please help

 

[Samy_A]

View attachment 96332
This was as far as I could go
I'm wondering how that 1/a came and how to make this into a 1/x^2+1 formate so I can input tan
Please help

You can do it in two (very similar) ways.
I assume that you found the indefinite integral $\int \frac{1}{1+x²}dx$.
To calculate the indefinite integral $\int \frac{1}{a²+x²}dx$, you could:
1) use the substitution $x=a \tan y$ and solve the same way as you did for $\int \frac{1}{1+x²}dx$;
2) use the substitution $x=ay$, which gives $\int \frac{1}{a²+x²}dx =\int \frac{1}{a²+a²y²} ady =\frac{1}{a} \int \frac{1}{1+y²} dy$, the indefinite integral you already solved (up to a constant 1/a).

Just to be clear, all my integrals here are indefinite integrals. When you calculate your definite integral, watch the integration limits when you perform a substitution.

 

[lioric]

I finally figured it out

It can be taken like this
It's sort of like the perfect square rule this once I put it to this formate it's done
Thank you very much