1 / | | 3 | sin (x)*x*cos(x) dx | / 0
Integral(sin(x)^3*x*cos(x), (x, 0, 1))
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 is when :
Now substitute back in:
Rewrite the integrand:
Let .
Then let and substitute :
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 is the constant times the variable of integration:
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:
Rewrite the integrand:
There are multiple ways to do this integral.
Rewrite the integrand:
Integrate term-by-term:
The integral of a constant times a function is the constant times the integral of the function:
Rewrite the integrand:
Integrate term-by-term:
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 integral of a constant is the constant times the variable of integration:
The result is:
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 integral of a constant is the constant times the variable of integration:
The result is:
Rewrite the integrand:
Integrate term-by-term:
The integral of a constant times a function is the constant times the integral of the function:
Rewrite the integrand:
Integrate term-by-term:
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 integral of a constant is the constant times the variable of integration:
The result is:
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 integral of a constant is the constant times the variable of integration:
The result is:
So, the result is:
Add the constant of integration:
The answer is:
/ | 4 | 3 3*x sin(4*x) sin(2*x) x*sin (x) | sin (x)*x*cos(x) dx = C - --- - -------- + -------- + --------- | 32 128 16 4 /
4 4 2 2 3 3 3*cos (1) 5*sin (1) 3*cos (1)*sin (1) 3*cos (1)*sin(1) 5*sin (1)*cos(1) - --------- + --------- - ----------------- + ---------------- + ---------------- 32 32 16 32 32
=
4 4 2 2 3 3 3*cos (1) 5*sin (1) 3*cos (1)*sin (1) 3*cos (1)*sin(1) 5*sin (1)*cos(1) - --------- + --------- - ----------------- + ---------------- + ---------------- 32 32 16 32 32
Use the examples entering the upper and lower limits of integration.