pi -- 3 / | | 3 | cos (x) | ------- dx | sin(x) | / pi -- 4
Integral(cos(x)^3/sin(x), (x, pi/4, pi/3))
Rewrite the integrand:
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:
Let .
Then let and substitute :
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 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 .
So, the result is:
The result is:
So, the result is:
Now substitute back in:
So, the result is:
Now substitute back in:
Rewrite the integrand:
The integral of a constant times a function is the constant times the integral of the function:
Let .
Then let and substitute :
Let .
Then let and substitute :
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 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 .
So, the result is:
The result is:
So, the result is:
Now substitute back in:
Now substitute back in:
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:
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 is .
Now substitute back in:
The result is:
Add the constant of integration:
The answer is:
/ | | 3 / 2 \ 2 | cos (x) log\sin (x)/ sin (x) | ------- dx = C + ------------ - ------- | sin(x) 2 2 | /
/ ___\ / ___\ 1 |\/ 2 | |\/ 3 | - - - log|-----| + log|-----| 8 \ 2 / \ 2 /
=
/ ___\ / ___\ 1 |\/ 2 | |\/ 3 | - - - log|-----| + log|-----| 8 \ 2 / \ 2 /
-1/8 - log(sqrt(2)/2) + log(sqrt(3)/2)
Use the examples entering the upper and lower limits of integration.