x / | | 3 | log (t) dt | / 2
Integral(log(t)^3, (t, 2, x))
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.
Use integration by parts:
Let and let .
Then .
To find :
The integral of the exponential function is itself.
Now evaluate the sub-integral.
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 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 simplify:
Add the constant of integration:
The answer is:
/ | | 3 3 2 | log (t) dt = C - 6*t + t*log (t) - 3*t*log (t) + 6*t*log(t) | /
3 2 3 2 12 - 12*log(2) - 6*x - 2*log (2) + 6*log (2) + x*log (x) - 3*x*log (x) + 6*x*log(x)
=
3 2 3 2 12 - 12*log(2) - 6*x - 2*log (2) + 6*log (2) + x*log (x) - 3*x*log (x) + 6*x*log(x)
Use the examples entering the upper and lower limits of integration.