How to Integrate 1/sqrt(1+x^2) dx | Step-by-Step Solution

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Homework Help Overview

The discussion revolves around the integral of 1/sqrt(1+x^2) dx, which falls under the subject area of calculus, specifically integration techniques.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the substitution of x=tan(t) and the resulting integral transformation. There is a mention of different approaches yielding varying answers, particularly the involvement of sine in one of the solutions. Questions arise regarding the correctness of expressing the integral in terms of arctan.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the integral and questioning the validity of certain expressions. Some guidance is offered regarding integration formulas, but no consensus has been reached on the correct approach.

Contextual Notes

Participants note potential confusion stemming from differing methods and results, as well as the implications of using specific substitutions in the integration process.

Mathematicsss

Homework Statement


Integral of 1/sqrt(1+x^2) dx

Homework Equations


sin^2theta`+cos^2theta=1
1+tan^2theta=sec^2theta

The Attempt at a Solution


I plugged x=tant --> dx=sec^2t dt
=> integral of 1/sqrt9(1+tan^2t) sec^2t dt
= integral of t = tan^-1t + C

However, another answer I've seen involves sin, why is that?
 
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Mathematicsss said:

Homework Statement


Integral of 1/sqrt(1+x^2) dx

Homework Equations


sin^2theta`+cos^2theta=1
1+tan^2theta=sec^2theta

The Attempt at a Solution


I plugged x=tant --> dx=sec^2t dt
=> integral of 1/sqrt9(1+tan^2t) sec^2t dt
= integral of t = tan^-1t + C

However, another answer I've seen involves sin, why is that?

Setting ##t = \arctan(t) + C## is wrong.
 
Ray Vickson said:
Setting ##t = \arctan(t) + C## is wrong.
I meant t=arctan(x)+C
 
Hello

This is a formula.
Check Integration Formulas - Integration of Special Functions, number 6
 

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