1 / | | sin(2*x)*sin(x)*cos(3*x) dx | / 0
Integral(sin(2*x)*sin(x)*cos(3*x), (x, 0, 1))
There are multiple ways to do this integral.
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:
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:
Rewrite the integrand:
Let .
Then let and substitute :
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:
The integral of is when :
So, the result is:
The result is:
Now substitute back in:
So, the result is:
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:
The integral of cosine is sine:
So, the result is:
The integral of a constant is the constant times the variable of integration:
The result is:
Now substitute back in:
So, the result is:
The integral of a constant times a function is the constant times the integral of the function:
Rewrite the integrand:
Let .
Then let and substitute :
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:
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 result is:
Now substitute back in:
So, the result is:
The result is:
So, 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:
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:
Rewrite the integrand:
Let .
Then let and substitute :
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:
The integral of is when :
So, the result is:
The result is:
Now substitute back in:
So, the result is:
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:
The integral of cosine is sine:
So, the result is:
The integral of a constant is the constant times the variable of integration:
The result is:
Now substitute back in:
So, the result is:
The integral of a constant times a function is the constant times the integral of the function:
Rewrite the integrand:
Let .
Then let and substitute :
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:
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 result is:
Now substitute back in:
So, the result is:
The result is:
Add the constant of integration:
The answer is:
/ 3 | x sin (2*x) sin(4*x) | sin(2*x)*sin(x)*cos(3*x) dx = C - - + --------- + -------- | 4 6 16 /
cos(1)*cos(2)*cos(3) cos(1)*sin(2)*sin(3) cos(2)*sin(1)*sin(3) cos(1)*cos(3)*sin(2) cos(3)*sin(1)*sin(2) cos(1)*cos(2)*sin(3) 5*sin(1)*sin(2)*sin(3) - -------------------- - -------------------- - -------------------- - -------------------- + -------------------- + -------------------- + ---------------------- 4 4 4 8 4 6 24
=
cos(1)*cos(2)*cos(3) cos(1)*sin(2)*sin(3) cos(2)*sin(1)*sin(3) cos(1)*cos(3)*sin(2) cos(3)*sin(1)*sin(2) cos(1)*cos(2)*sin(3) 5*sin(1)*sin(2)*sin(3) - -------------------- - -------------------- - -------------------- - -------------------- + -------------------- + -------------------- + ---------------------- 4 4 4 8 4 6 24
Use the examples entering the upper and lower limits of integration.