1 / | | ___ 2 | \/ x *log (x) dx | / 0
Integral(sqrt(x)*log(x)^2, (x, 0, 1))
Let .
Then let and substitute :
Use integration by parts:
Let and let .
Then .
To find :
There are multiple ways to do this integral.
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:
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 a constant is the constant times the variable of integration:
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:
Now simplify:
Add the constant of integration:
The answer is:
/ | 3/2 3/2 3/2 2 | ___ 2 16*x 8*x *log(x) 2*x *log (x) | \/ x *log (x) dx = C + ------- - ------------- + -------------- | 27 9 3 /
1 / | | / /1 \ | | 0 for And|- < 1, x < 1| | | \x / | | | | ___ 2 /1 \ | | \/ x *log (x) for Or|- < 1, x < 1| | | \x / | < dx | | / __0, 4 /5/2, 5/2, 5/2, 1 | \ \ | | |3*/__ | | x| | | | | \_|4, 4 \ 3/2, 3/2, 3/2, 0 | / __0, 4 /3/2, 5/2, 5/2, 1 | \| __3, 1 / 0 5/2, 5/2, 5/2 | \ | |2*|-------------------------------------------------- + /__ | | x|| 2*/__ | | x| | | \ 2 \_|4, 4 \ 3/2, 3/2, 3/2, 0 | // \_|4, 4 \3/2, 3/2, 3/2 0 | / | |--------------------------------------------------------------------------------------------------------- + -------------------------------------------- otherwise | \ x x | / 0
=
1 / | | / /1 \ | | 0 for And|- < 1, x < 1| | | \x / | | | | ___ 2 /1 \ | | \/ x *log (x) for Or|- < 1, x < 1| | | \x / | < dx | | / __0, 4 /5/2, 5/2, 5/2, 1 | \ \ | | |3*/__ | | x| | | | | \_|4, 4 \ 3/2, 3/2, 3/2, 0 | / __0, 4 /3/2, 5/2, 5/2, 1 | \| __3, 1 / 0 5/2, 5/2, 5/2 | \ | |2*|-------------------------------------------------- + /__ | | x|| 2*/__ | | x| | | \ 2 \_|4, 4 \ 3/2, 3/2, 3/2, 0 | // \_|4, 4 \3/2, 3/2, 3/2 0 | / | |--------------------------------------------------------------------------------------------------------- + -------------------------------------------- otherwise | \ x x | / 0
Use the examples entering the upper and lower limits of integration.