1 / | | log(2*x) | ---------- dx | x*log(4*x) | / 0
Integral(log(2*x)/((x*log(4*x))), (x, 0, 1))
There are multiple ways to do this integral.
Rewrite the integrand:
Let .
Then let and substitute :
Rewrite the integrand:
Integrate term-by-term:
The integral of a constant is the constant times the variable of integration:
The integral of a constant times a function is the constant times the integral of the function:
Let .
Then let and substitute :
The integral of is .
Now substitute back in:
So, the result is:
The result is:
Now substitute back in:
Use integration by parts:
Let and let .
Then .
To find :
Let .
Then let and substitute :
The integral of is .
Now substitute back in:
Now evaluate the sub-integral.
Let .
Then let and substitute :
The integral of a constant times a function is the constant times the integral of the function:
Let .
Then let and substitute :
The integral of a constant times a function is the constant times the integral of the function:
Let .
Then let and substitute :
Use integration by parts:
Let and let .
Then .
To find :
The integral of the exponential function is itself.
Now evaluate the sub-integral.
The integral of the exponential function is itself.
Now substitute back in:
So, the result is:
Now substitute back in:
So, the result is:
Now substitute back in:
Now simplify:
Add the constant of integration:
The answer is:
/ | | log(2*x) | ---------- dx = C - log(2)*log(2*log(2) + log(x)) + log(x) | x*log(4*x) | /
Use the examples entering the upper and lower limits of integration.