# Integral of log2x dx

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### The solution

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|  log(2*x) dx
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$$\int\limits_{0}^{1} \log{\left(2 x \right)}\, dx$$
Detail solution
1. There are multiple ways to do this integral.

## Method #1

1. Let .

Then let and substitute :

1. The integral of a constant times a function is the constant times the integral of the function:

1. Use integration by parts:

Let and let .

Then .

To find :

1. The integral of a constant is the constant times the variable of integration:

Now evaluate the sub-integral.

2. The integral of a constant is the constant times the variable of integration:

So, the result is:

Now substitute back in:

## Method #2

1. Use integration by parts:

Let and let .

Then .

To find :

1. The integral of a constant is the constant times the variable of integration:

Now evaluate the sub-integral.

2. The integral of a constant is the constant times the variable of integration:

2. Now simplify:

3. Add the constant of integration:

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| log(2*x) dx = C - x + x*log(2*x)
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$${{2\,x\,\log \left(2\,x\right)-2\,x}\over{2}}$$
-1 + log(2)
$${{2\,\log 2-2}\over{2}}$$
=
-1 + log(2)
$$-1 + \log{\left(2 \right)}$$
-0.306852819440055
-0.306852819440055