1 / | | /x - pi\ | x*cos|------| dx | \ 3 / | / 0
Integral(x*cos((x - pi)/3), (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 sine is negative cosine:
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 cosine is sine:
So, the result is:
Now substitute back in:
So, the result is:
Add the constant of integration:
The answer is:
/ | | /x - pi\ /x pi\ /x pi\ | x*cos|------| dx = C + 9*sin|- + --| - 3*x*cos|- + --| | \ 3 / \3 6 / \3 6 / | /
9 /1 pi\ /1 pi\ - - - 3*cos|- + --| + 9*sin|- + --| 2 \3 6 / \3 6 /
=
9 /1 pi\ /1 pi\ - - - 3*cos|- + --| + 9*sin|- + --| 2 \3 6 / \3 6 /
-9/2 - 3*cos(1/3 + pi/6) + 9*sin(1/3 + pi/6)
Use the examples entering the upper and lower limits of integration.