E / | | 2 | x *log(2*x) dx | / 1
Integral(x^2*log(2*x), (x, 1, E))
There are multiple ways to do this integral.
Rewrite the integrand:
Integrate term-by-term:
Let .
Then let and substitute :
Use integration by parts:
Let and let .
Then .
To find :
Let .
Then let and substitute :
The integral of a constant times a function is the constant times the integral of the function:
The integral of the exponential function is itself.
So, the result is:
Now substitute back in:
Now evaluate the sub-integral.
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:
The integral of the exponential function is itself.
So, the result is:
Now substitute back in:
So, the result is:
Now substitute back in:
The integral of a constant times a function is the constant times the integral of the function:
The integral of is when :
So, the result is:
The result is:
Use integration by parts:
Let and let .
Then .
To find :
The integral of is when :
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:
The integral of is when :
So, the result is:
Rewrite the integrand:
Integrate term-by-term:
Let .
Then let and substitute :
Use integration by parts:
Let and let .
Then .
To find :
Let .
Then let and substitute :
The integral of a constant times a function is the constant times the integral of the function:
The integral of the exponential function is itself.
So, the result is:
Now substitute back in:
Now evaluate the sub-integral.
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:
The integral of the exponential function is itself.
So, the result is:
Now substitute back in:
So, the result is:
Now substitute back in:
The integral of a constant times a function is the constant times the integral of the function:
The integral of is when :
So, the result is:
The result is:
Now simplify:
Add the constant of integration:
The answer is:
/ | 3 3 3 | 2 x x *log(2) x *log(x) | x *log(2*x) dx = C - -- + --------- + --------- | 9 3 3 /
3 3 1 log(2) e e *log(2*E) - - ------ - -- + ----------- 9 3 9 3
=
3 3 1 log(2) e e *log(2*E) - - ------ - -- + ----------- 9 3 9 3
1/9 - log(2)/3 - exp(3)/9 + exp(3)*log(2*E)/3
Use the examples entering the upper and lower limits of integration.