3 / | | / 2 \ | \x*(1 - log(x)) + 1/ dx | / 1
Integral(x*(1 - log(x))^2 + 1, (x, 1, 3))
Integrate term-by-term:
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.
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:
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:
So, the result is:
The integral of is when :
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.
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:
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:
So, the result is:
The integral of is when :
The result is:
The integral of a constant is the constant times the variable of integration:
The result is:
Now simplify:
Add the constant of integration:
The answer is:
/ | 2 2 2 2 | / 2 \ 5*x x *log (x) 3*x *log(x) | \x*(1 - log(x)) + 1/ dx = C + x + ---- + ---------- - ----------- | 4 2 2 /
2 27*log(3) 9*log (3) 12 - --------- + --------- 2 2
=
2 27*log(3) 9*log (3) 12 - --------- + --------- 2 2
12 - 27*log(3)/2 + 9*log(3)^2/2
Use the examples entering the upper and lower limits of integration.