Ikunupe6v

2021-12-31

Demystify integration of $\int \frac{1}{x}dx$

Ive

Ive

Marcus Herman

Beginner2022-01-01Added 41 answers

If you want to try to prove $\int \frac{dx}{x}=\mathrm{ln}x+C\left(\text{}\text{for 0}\right)$ , try the substitution

$x={e}^{u}$

$dx={e}^{u}du$

This substitution is justified because the exponential function is bijective from$\mathbb{R}$ to $(0,\mathrm{\infty})$ (hence for every x there exists a u) and continuously differentiable (which allows an integration by substitution).

$\int \frac{dx}{x}=\int \frac{{e}^{u}du}{{e}^{u}}=u+C$

Now just use the fact that natural log is the inverse of the exponential function. If

$x={e}^{u},u=\mathrm{ln}x$

This substitution is justified because the exponential function is bijective from

Now just use the fact that natural log is the inverse of the exponential function. If

Ronnie Schechter

Beginner2022-01-02Added 27 answers

Lets

Vasquez

Expert2022-01-09Added 669 answers

To show

and the definition of the derivative. We have

which is what we wanted to show.

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