1 / | | 5 4 | sin (x)*cos (x) dx | / 0
Integral(sin(x)^5*cos(x)^4, (x, 0, 1))
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 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:
The integral of is when :
So, the result is:
The result is:
Now substitute back in:
Rewrite the integrand:
Integrate term-by-term:
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:
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 is when :
So, the result is:
Now substitute back in:
The result is:
Rewrite the integrand:
Integrate term-by-term:
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:
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 is when :
So, the result is:
Now substitute back in:
The result is:
Now simplify:
Add the constant of integration:
The answer is:
/ | 5 9 7 | 5 4 cos (x) cos (x) 2*cos (x) | sin (x)*cos (x) dx = C - ------- - ------- + --------- | 5 9 7 /
5 9 7 8 cos (1) cos (1) 2*cos (1) --- - ------- - ------- + --------- 315 5 9 7
=
5 9 7 8 cos (1) cos (1) 2*cos (1) --- - ------- - ------- + --------- 315 5 9 7
8/315 - cos(1)^5/5 - cos(1)^9/9 + 2*cos(1)^7/7
Use the examples entering the upper and lower limits of integration.