2*p / | | 3/x\ | sin |-| dx | \4/ | / 0
Integral(sin(x/4)^3, (x, 0, 2*p))
Rewrite the integrand:
There are multiple ways to do this integral.
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:
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 is when :
So, the result is:
Now substitute back in:
So, the result is:
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 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:
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:
So, the result is:
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 result is:
Now simplify:
Add the constant of integration:
The answer is:
/ 3/x\ | 4*cos |-| | 3/x\ /x\ \4/ | sin |-| dx = C - 4*cos|-| + --------- | \4/ \4/ 3 | /
3/p\ 4*cos |-| 8 /p\ \2/ - - 4*cos|-| + --------- 3 \2/ 3
=
3/p\ 4*cos |-| 8 /p\ \2/ - - 4*cos|-| + --------- 3 \2/ 3
8/3 - 4*cos(p/2) + 4*cos(p/2)^3/3
Use the examples entering the upper and lower limits of integration.