Lillianna Andersen

2022-07-03

Two apparently different antiderivatives of $\frac{1}{2x}$

What is right way to calculate this integral and why?

$\int \frac{1}{2x}\text{d}x$

I thought, that this substitution is right:

$t=2x$

$\text{d}t=2\text{d}x$

$\frac{\text{d}t}{2}=\text{d}x$

$\int \frac{1}{2x}\text{d}x=\int \frac{1}{t}\frac{\text{d}t}{2}=\frac{1}{2}\mathrm{ln}|2x|+C.$

But it's not right, because this is the correct answer:

$\int \frac{1}{2x}\text{d}x=\frac{1}{2}\int \frac{1}{x}\text{d}x=\frac{1}{2}\mathrm{ln}|x|+C.$

Can someone explain me, why is the first way wrong?

What is right way to calculate this integral and why?

$\int \frac{1}{2x}\text{d}x$

I thought, that this substitution is right:

$t=2x$

$\text{d}t=2\text{d}x$

$\frac{\text{d}t}{2}=\text{d}x$

$\int \frac{1}{2x}\text{d}x=\int \frac{1}{t}\frac{\text{d}t}{2}=\frac{1}{2}\mathrm{ln}|2x|+C.$

But it's not right, because this is the correct answer:

$\int \frac{1}{2x}\text{d}x=\frac{1}{2}\int \frac{1}{x}\text{d}x=\frac{1}{2}\mathrm{ln}|x|+C.$

Can someone explain me, why is the first way wrong?

Tamia Padilla

Beginner2022-07-04Added 16 answers

Step 1

The two answers actually agree, that is, the first calculation really does produce an antiderivative of $\frac{1}{2x}$ as expected, at least provided we interpret C to be a general constant in both cases. Expanding the first antiderivative gives $\frac{1}{2}\mathrm{ln}|2x|+C=\frac{1}{2}(\mathrm{log}|x|+\mathrm{log}2)+C=\frac{1}{2}\mathrm{log}|x|+(C+\frac{1}{2}\mathrm{log}2).$

Step 2

So, if we denote ${C}^{\prime}:=C+\frac{1}{2}\mathrm{log}2$, this antiderivative is $\frac{1}{2}\mathrm{ln}|x|+{C}^{\prime},$, which (again, regarded as a family of functions, all equal up to an over constant) coincides with the second antiderivative.

This exemplifies the statement that an antiderivative of a given function is unique up to addition of an overall constant. (Actually, this only need be true when the domain of the function is connected, which in particular is not the case for the given integrand, but this isn't essential to the question at hand.)

The two answers actually agree, that is, the first calculation really does produce an antiderivative of $\frac{1}{2x}$ as expected, at least provided we interpret C to be a general constant in both cases. Expanding the first antiderivative gives $\frac{1}{2}\mathrm{ln}|2x|+C=\frac{1}{2}(\mathrm{log}|x|+\mathrm{log}2)+C=\frac{1}{2}\mathrm{log}|x|+(C+\frac{1}{2}\mathrm{log}2).$

Step 2

So, if we denote ${C}^{\prime}:=C+\frac{1}{2}\mathrm{log}2$, this antiderivative is $\frac{1}{2}\mathrm{ln}|x|+{C}^{\prime},$, which (again, regarded as a family of functions, all equal up to an over constant) coincides with the second antiderivative.

This exemplifies the statement that an antiderivative of a given function is unique up to addition of an overall constant. (Actually, this only need be true when the domain of the function is connected, which in particular is not the case for the given integrand, but this isn't essential to the question at hand.)

Sylvia Byrd

Beginner2022-07-05Added 6 answers

Explanation:

You are correct.

Since $\mathrm{ln}|2x|=\mathrm{ln}|x|+\mathrm{ln}2$

$\int \frac{1}{2x}\text{}dx=\mathrm{ln}|2x|+{C}_{1}=\mathrm{ln}|x|+\stackrel{\text{New Const}}{\stackrel{\u23de}{\mathrm{ln}2+{C}_{1}}}=\mathrm{ln}|x|+C$

You are correct.

Since $\mathrm{ln}|2x|=\mathrm{ln}|x|+\mathrm{ln}2$

$\int \frac{1}{2x}\text{}dx=\mathrm{ln}|2x|+{C}_{1}=\mathrm{ln}|x|+\stackrel{\text{New Const}}{\stackrel{\u23de}{\mathrm{ln}2+{C}_{1}}}=\mathrm{ln}|x|+C$

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

$f(x,y)={x}^{3}-6xy+8{y}^{3}$ $\frac{1}{\mathrm{sec}(x)}$ in derivative?

What is the derivative of $\mathrm{ln}(x+1)$?

What is the limit of $e}^{-x$ as $x\to \infty$?

What is the derivative of $f\left(x\right)={5}^{\mathrm{ln}x}$?

What is the derivative of $e}^{-2x$?

How to find $lim\frac{{e}^{t}-1}{t}$ as $t\to 0$ using l'Hospital's Rule?

What is the integral of $\sqrt{9-{x}^{2}}$?

What is the derivative of $f\left(x\right)=\mathrm{ln}\left[{x}^{9}{(x+3)}^{6}{({x}^{2}+7)}^{5}\right]$ ?

What Is the common difference or common ratio of the sequence 2, 5, 8, 11...?

How to find the derivative of $y={e}^{5x}$?

How to evaluate the limit $\frac{\mathrm{sin}\left(5x\right)}{x}$ as x approaches 0?

How to find derivatives of parametric functions?

What is the antiderivative of $-5{e}^{x-1}$?

How to evaluate: indefinite integral $\frac{1+x}{1+{x}^{2}}dx$?