1 / | | 2 2 | x *sin (x) dx | / 0
Integral(x^2*sin(x)^2, (x, 0, 1))
Use integration by parts:
Let and let .
Then .
To find :
Rewrite the integrand:
Integrate term-by-term:
The integral of a constant is the constant times the variable of integration:
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:
The result is:
Now evaluate the sub-integral.
Use integration by parts:
Let and let .
Then .
To find :
Integrate term-by-term:
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:
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 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:
So, the result is:
The result is:
Now evaluate the sub-integral.
Integrate term-by-term:
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:
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:
The result is:
Now simplify:
Add the constant of integration:
The answer is:
/ / 2 cos(2*x)\ | 3 x*|x + --------| | 2 2 x sin(2*x) 2 /x sin(2*x)\ \ 2 / | x *sin (x) dx = C + -- + -------- + x *|- - --------| - ----------------- | 6 8 \2 4 / 2 /
2 2 cos (1) 5*sin (1) cos(1)*sin(1) - ------- + --------- - ------------- 12 12 4
=
2 2 cos (1) 5*sin (1) cos(1)*sin(1) - ------- + --------- - ------------- 12 12 4
Use the examples entering the upper and lower limits of integration.