What Is the Correct Substitution for the Integral of 4x/(1+4x^2)?

[sparkle123]

How do you get that the integral of 4x/(1+4x^2) = 1/2*ln(1_4x^2) ?
I have no clue. I tried u substitution with y=4x^2 but that got me nowhere.
Thanks!

 

[Dick]

Try substitution with y=1+4x^2.

 

[hunt_mat]

Note that:

$$\frac{4x}{1+4x^{2}}=\frac{1}{2}\frac{8x}{1+4x^{2}}=\frac{1}{2}\frac{f'(x)}{f(x)}$$

Where $$f(x)=1+4x^{2}$$

 

[sparkle123]

Thanks hunt_mat and Dick! :)

 

[dextercioby]

How do you get that the integral of 4x/(1+4x^2) = 1/2*ln(1_4x^2) ?
I have no clue. I tried u substitution with y=4x^2 but that got me nowhere.
Thanks!

Even though you could have included the 1 in the substitution, the one you chose would have solved your problem the same way since

$$\int \frac{du}{1+u} = \ln (1+u) + C$$ , whenever u>0.

 

[sparkle123]

Cool thanks!