0 / | | 5 | sin (x) dx | / -1
Integral(sin(x)^5, (x, -1, 0))
Rewrite the integrand:
There are multiple ways to do this integral.
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:
The integral of sine is negative cosine:
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:
The integral of sine is negative cosine:
The result is:
Add the constant of integration:
The answer is:
/ | 5 3 | 5 cos (x) 2*cos (x) | sin (x) dx = C - cos(x) - ------- + --------- | 5 3 /
3 5 8 2*cos (1) cos (1) - -- - --------- + ------- + cos(1) 15 3 5
=
3 5 8 2*cos (1) cos (1) - -- - --------- + ------- + cos(1) 15 3 5
-8/15 - 2*cos(1)^3/3 + cos(1)^5/5 + cos(1)
Use the examples entering the upper and lower limits of integration.