1 / | | 2 | x *cos(2*x) dx | / 0
Integral(x^2*cos(2*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.
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 sine is negative cosine:
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:
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 is when :
So, the result is:
Now substitute back in:
Let .
Then let and substitute :
The integral of is when :
Now substitute back in:
So, the result is:
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:
/ | 2 | 2 sin(2*x) x*cos(2*x) x *sin(2*x) | x *cos(2*x) dx = C - -------- + ---------- + ----------- | 4 2 2 /
cos(2) sin(2) ------ + ------ 2 4
=
cos(2) sin(2) ------ + ------ 2 4
cos(2)/2 + sin(2)/4
Use the examples entering the upper and lower limits of integration.