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

[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?

 

[Ray Vickson]

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.

 

[Mathematicsss]

Setting $t = \arctan(t) + C$ is wrong.

I meant t=arctan(x)+C

 

[Zack554]

Hello

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