1 / | | x*cos(pi*x) dx | / 0
Integral(x*cos(pi*x), (x, 0, 1))
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 cosine is sine:
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 sine is negative cosine:
So, the result is:
Now substitute back in:
So, the result is:
Now simplify:
Add the constant of integration:
The answer is:
/ | cos(pi*x) x*sin(pi*x) | x*cos(pi*x) dx = C + --------- + ----------- | 2 pi / pi
-2 --- 2 pi
=
-2 --- 2 pi
-2/pi^2
Use the examples entering the upper and lower limits of integration.